diff --git a/src/accessibility/RovingTabIndex.tsx b/src/accessibility/RovingTabIndex.tsx
index 87f525bdfc..4e2279e09f 100644
--- a/src/accessibility/RovingTabIndex.tsx
+++ b/src/accessibility/RovingTabIndex.tsx
@@ -150,13 +150,14 @@ const reducer = (state: IState, action: IAction) => {
 
 interface IProps {
     handleHomeEnd?: boolean;
+    handleUpDown?: boolean;
     children(renderProps: {
         onKeyDownHandler(ev: React.KeyboardEvent);
     });
     onKeyDown?(ev: React.KeyboardEvent, state: IState);
 }
 
-export const RovingTabIndexProvider: React.FC<IProps> = ({ children, handleHomeEnd, onKeyDown }) => {
+export const RovingTabIndexProvider: React.FC<IProps> = ({ children, handleHomeEnd, handleUpDown, onKeyDown }) => {
     const [state, dispatch] = useReducer<Reducer<IState, IAction>>(reducer, {
         activeRef: null,
         refs: [],
@@ -167,21 +168,50 @@ export const RovingTabIndexProvider: React.FC<IProps> = ({ children, handleHomeE
     const onKeyDownHandler = useCallback((ev) => {
         let handled = false;
         // Don't interfere with input default keydown behaviour
-        if (handleHomeEnd && ev.target.tagName !== "INPUT" && ev.target.tagName !== "TEXTAREA") {
+        if (ev.target.tagName !== "INPUT" && ev.target.tagName !== "TEXTAREA") {
             // check if we actually have any items
             switch (ev.key) {
                 case Key.HOME:
-                    handled = true;
-                    // move focus to first item
-                    if (context.state.refs.length > 0) {
-                        context.state.refs[0].current.focus();
+                    if (handleHomeEnd) {
+                        handled = true;
+                        // move focus to first item
+                        if (context.state.refs.length > 0) {
+                            context.state.refs[0].current.focus();
+                        }
                     }
                     break;
+
                 case Key.END:
-                    handled = true;
-                    // move focus to last item
-                    if (context.state.refs.length > 0) {
-                        context.state.refs[context.state.refs.length - 1].current.focus();
+                    if (handleHomeEnd) {
+                        handled = true;
+                        // move focus to last item
+                        if (context.state.refs.length > 0) {
+                            context.state.refs[context.state.refs.length - 1].current.focus();
+                        }
+                    }
+                    break;
+
+                case Key.ARROW_UP:
+                    if (handleUpDown) {
+                        handled = true;
+                        if (context.state.refs.length > 0) {
+                            const idx = context.state.refs.indexOf(context.state.activeRef);
+                            if (idx > 1) {
+                                context.state.refs[idx - 1].current.focus();
+                            }
+                        }
+                    }
+                    break;
+
+                case Key.ARROW_DOWN:
+                    if (handleUpDown) {
+                        handled = true;
+                        if (context.state.refs.length > 0) {
+                            const idx = context.state.refs.indexOf(context.state.activeRef);
+                            if (idx < context.state.refs.length - 1) {
+                                context.state.refs[idx + 1].current.focus();
+                            }
+                        }
                     }
                     break;
             }
@@ -193,7 +223,7 @@ export const RovingTabIndexProvider: React.FC<IProps> = ({ children, handleHomeE
         } else if (onKeyDown) {
             return onKeyDown(ev, context.state);
         }
-    }, [context.state, onKeyDown, handleHomeEnd]);
+    }, [context.state, onKeyDown, handleHomeEnd, handleUpDown]);
 
     return <RovingTabIndexContext.Provider value={context}>
         { children({ onKeyDownHandler }) }
diff --git a/src/components/structures/SpaceRoomDirectory.tsx b/src/components/structures/SpaceRoomDirectory.tsx
index d8cc9593f0..b2d5e787f5 100644
--- a/src/components/structures/SpaceRoomDirectory.tsx
+++ b/src/components/structures/SpaceRoomDirectory.tsx
@@ -14,7 +14,7 @@ See the License for the specific language governing permissions and
 limitations under the License.
 */
 
-import React, { ReactNode, useMemo, useState } from "react";
+import React, { ReactNode, KeyboardEvent, useMemo, useState } from "react";
 import { Room } from "matrix-js-sdk/src/models/room";
 import { EventType, RoomType } from "matrix-js-sdk/src/@types/event";
 import { ISpaceSummaryRoom, ISpaceSummaryEvent } from "matrix-js-sdk/src/@types/spaces";
@@ -46,6 +46,8 @@ import { getDisplayAliasForAliasSet } from "../../Rooms";
 import { useDispatcher } from "../../hooks/useDispatcher";
 import defaultDispatcher from "../../dispatcher/dispatcher";
 import { Action } from "../../dispatcher/actions";
+import { Key } from "../../Keyboard";
+import { IState, RovingTabIndexProvider, useRovingTabIndex } from "../../accessibility/RovingTabIndex";
 
 interface IHierarchyProps {
     space: Room;
@@ -80,6 +82,7 @@ const Tile: React.FC<ITileProps> = ({
         || (room.room_type === RoomType.Space ? _t("Unnamed Space") : _t("Unnamed Room"));
 
     const [showChildren, toggleShowChildren] = useStateToggle(true);
+    const [onFocus, isActive, ref] = useRovingTabIndex();
 
     const onPreviewClick = (ev: ButtonEvent) => {
         ev.preventDefault();
@@ -94,11 +97,21 @@ const Tile: React.FC<ITileProps> = ({
 
     let button;
     if (joinedRoom) {
-        button = <AccessibleButton onClick={onPreviewClick} kind="primary_outline">
+        button = <AccessibleButton
+            onClick={onPreviewClick}
+            kind="primary_outline"
+            onFocus={onFocus}
+            tabIndex={isActive ? 0 : -1}
+        >
             { _t("View") }
         </AccessibleButton>;
     } else if (onJoinClick) {
-        button = <AccessibleButton onClick={onJoinClick} kind="primary">
+        button = <AccessibleButton
+            onClick={onJoinClick}
+            kind="primary"
+            onFocus={onFocus}
+            tabIndex={isActive ? 0 : -1}
+        >
             { _t("Join") }
         </AccessibleButton>;
     }
@@ -106,13 +119,13 @@ const Tile: React.FC<ITileProps> = ({
     let checkbox;
     if (onToggleClick) {
         if (hasPermissions) {
-            checkbox = <StyledCheckbox checked={!!selected} onChange={onToggleClick} />;
+            checkbox = <StyledCheckbox checked={!!selected} onChange={onToggleClick} tabIndex={isActive ? 0 : -1} />;
         } else {
             checkbox = <TextWithTooltip
                 tooltip={_t("You don't have permission")}
                 onClick={ev => { ev.stopPropagation(); }}
             >
-                <StyledCheckbox disabled={true} />
+                <StyledCheckbox disabled={true} tabIndex={isActive ? 0 : -1} />
             </TextWithTooltip>;
         }
     }
@@ -185,19 +198,65 @@ const Tile: React.FC<ITileProps> = ({
                 toggleShowChildren();
             }}
         />;
+
         if (showChildren) {
-            childSection = <div className="mx_SpaceRoomDirectory_subspace_children">
+            const onChildrenKeyDown = (e) => {
+                if (e.key === Key.ARROW_LEFT) {
+                    e.preventDefault();
+                    e.stopPropagation();
+                    ref.current?.focus();
+                }
+            };
+
+            childSection = <div
+                className="mx_SpaceRoomDirectory_subspace_children"
+                onKeyDown={onChildrenKeyDown}
+                role="group"
+            >
                 { children }
             </div>;
         }
     }
 
+    const onKeyDown = children ? (e) => {
+        let handled = false;
+
+        switch (e.key) {
+            case Key.ARROW_LEFT:
+                if (showChildren) {
+                    handled = true;
+                    toggleShowChildren();
+                }
+                break;
+
+            case Key.ARROW_RIGHT:
+                handled = true;
+                if (showChildren) {
+                    (ref.current?.nextElementSibling?.firstElementChild as HTMLElement)?.focus();
+                } else {
+                    toggleShowChildren();
+                }
+                break;
+        }
+
+        if (handled) {
+            e.preventDefault();
+            e.stopPropagation();
+        }
+    } : undefined;
+
     return <>
         <AccessibleButton
             className={classNames("mx_SpaceRoomDirectory_roomTile", {
                 mx_SpaceRoomDirectory_subspace: room.room_type === RoomType.Space,
             })}
             onClick={(hasPermissions && onToggleClick) ? onToggleClick : onPreviewClick}
+            onKeyDown={onKeyDown}
+            inputRef={ref}
+            onFocus={onFocus}
+            tabIndex={isActive ? 0 : -1}
+            aria-expanded={children ? showChildren : undefined}
+            role="treeitem"
         >
             { content }
             { childToggle }
@@ -414,176 +473,191 @@ export const SpaceHierarchy: React.FC<IHierarchyProps> = ({
         return <p>{ _t("Your server does not support showing space hierarchies.") }</p>;
     }
 
-    let content;
-    if (roomsMap) {
-        const numRooms = Array.from(roomsMap.values()).filter(r => !r.room_type).length;
-        const numSpaces = roomsMap.size - numRooms - 1; // -1 at the end to exclude the space we are looking at
-
-        let countsStr;
-        if (numSpaces > 1) {
-            countsStr = _t("%(count)s rooms and %(numSpaces)s spaces", { count: numRooms, numSpaces });
-        } else if (numSpaces > 0) {
-            countsStr = _t("%(count)s rooms and 1 space", { count: numRooms, numSpaces });
-        } else {
-            countsStr = _t("%(count)s rooms", { count: numRooms, numSpaces });
+    const onKeyDown = (ev: KeyboardEvent, state: IState) => {
+        if (ev.key === Key.ARROW_DOWN && ev.currentTarget.classList.contains("mx_SpaceRoomDirectory_search")) {
+            state.refs[0]?.current?.focus();
         }
-
-        let manageButtons;
-        if (space.getMyMembership() === "join" && space.currentState.maySendStateEvent(EventType.SpaceChild, userId)) {
-            const selectedRelations = Array.from(selected.keys()).flatMap(parentId => {
-                return [...selected.get(parentId).values()].map(childId => [parentId, childId]) as [string, string][];
-            });
-
-            const selectionAllSuggested = selectedRelations.every(([parentId, childId]) => {
-                return parentChildMap.get(parentId)?.get(childId)?.content.suggested;
-            });
-
-            const disabled = !selectedRelations.length || removing || saving;
-
-            let Button: React.ComponentType<React.ComponentProps<typeof AccessibleButton>> = AccessibleButton;
-            let props = {};
-            if (!selectedRelations.length) {
-                Button = AccessibleTooltipButton;
-                props = {
-                    tooltip: _t("Select a room below first"),
-                    yOffset: -40,
-                };
-            }
-
-            manageButtons = <>
-                <Button
-                    {...props}
-                    onClick={async () => {
-                        setRemoving(true);
-                        try {
-                            for (const [parentId, childId] of selectedRelations) {
-                                await cli.sendStateEvent(parentId, EventType.SpaceChild, {}, childId);
-                                parentChildMap.get(parentId).delete(childId);
-                                if (parentChildMap.get(parentId).size > 0) {
-                                    parentChildMap.set(parentId, new Map(parentChildMap.get(parentId)));
-                                } else {
-                                    parentChildMap.delete(parentId);
-                                }
-                            }
-                        } catch (e) {
-                            setError(_t("Failed to remove some rooms. Try again later"));
-                        }
-                        setRemoving(false);
-                    }}
-                    kind="danger_outline"
-                    disabled={disabled}
-                >
-                    { removing ? _t("Removing...") : _t("Remove") }
-                </Button>
-                <Button
-                    {...props}
-                    onClick={async () => {
-                        setSaving(true);
-                        try {
-                            for (const [parentId, childId] of selectedRelations) {
-                                const suggested = !selectionAllSuggested;
-                                const existingContent = parentChildMap.get(parentId)?.get(childId)?.content;
-                                if (!existingContent || existingContent.suggested === suggested) continue;
-
-                                const content = {
-                                    ...existingContent,
-                                    suggested: !selectionAllSuggested,
-                                };
-
-                                await cli.sendStateEvent(parentId, EventType.SpaceChild, content, childId);
-
-                                parentChildMap.get(parentId).get(childId).content = content;
-                                parentChildMap.set(parentId, new Map(parentChildMap.get(parentId)));
-                            }
-                        } catch (e) {
-                            setError("Failed to update some suggestions. Try again later");
-                        }
-                        setSaving(false);
-                        setSelected(new Map());
-                    }}
-                    kind="primary_outline"
-                    disabled={disabled}
-                >
-                    { saving
-                        ? _t("Saving...")
-                        : (selectionAllSuggested ? _t("Mark as not suggested") : _t("Mark as suggested"))
-                    }
-                </Button>
-            </>;
-        }
-
-        let results;
-        if (roomsMap.size) {
-            const hasPermissions = space?.currentState.maySendStateEvent(EventType.SpaceChild, cli.getUserId());
-
-            results = <>
-                <HierarchyLevel
-                    spaceId={space.roomId}
-                    rooms={roomsMap}
-                    relations={parentChildMap}
-                    parents={new Set()}
-                    selectedMap={selected}
-                    onToggleClick={hasPermissions ? (parentId, childId) => {
-                        setError("");
-                        if (!selected.has(parentId)) {
-                            setSelected(new Map(selected.set(parentId, new Set([childId]))));
-                            return;
-                        }
-
-                        const parentSet = selected.get(parentId);
-                        if (!parentSet.has(childId)) {
-                            setSelected(new Map(selected.set(parentId, new Set([...parentSet, childId]))));
-                            return;
-                        }
-
-                        parentSet.delete(childId);
-                        setSelected(new Map(selected.set(parentId, new Set(parentSet))));
-                    } : undefined}
-                    onViewRoomClick={(roomId, autoJoin) => {
-                        showRoom(roomsMap.get(roomId), Array.from(viaMap.get(roomId) || []), autoJoin);
-                    }}
-                />
-                { children && <hr /> }
-            </>;
-        } else {
-            results = <div className="mx_SpaceRoomDirectory_noResults">
-                <h3>{ _t("No results found") }</h3>
-                <div>{ _t("You may want to try a different search or check for typos.") }</div>
-            </div>;
-        }
-
-        content = <>
-            <div className="mx_SpaceRoomDirectory_listHeader">
-                { countsStr }
-                <span>
-                    { additionalButtons }
-                    { manageButtons }
-                </span>
-            </div>
-            { error && <div className="mx_SpaceRoomDirectory_error">
-                { error }
-            </div> }
-            <AutoHideScrollbar className="mx_SpaceRoomDirectory_list">
-                { results }
-                { children }
-            </AutoHideScrollbar>
-        </>;
-    } else {
-        content = <Spinner />;
-    }
+    };
 
     // TODO loading state/error state
-    return <>
-        <SearchBox
-            className="mx_textinput_icon mx_textinput_search"
-            placeholder={_t("Search names and descriptions")}
-            onSearch={setQuery}
-            autoFocus={true}
-            initialValue={initialText}
-        />
+    return <RovingTabIndexProvider onKeyDown={onKeyDown} handleHomeEnd handleUpDown>
+        { ({ onKeyDownHandler }) => {
+            let content;
+            if (roomsMap) {
+                const numRooms = Array.from(roomsMap.values()).filter(r => !r.room_type).length;
+                const numSpaces = roomsMap.size - numRooms - 1; // -1 at the end to exclude the space we are looking at
 
-        { content }
-    </>;
+                let countsStr;
+                if (numSpaces > 1) {
+                    countsStr = _t("%(count)s rooms and %(numSpaces)s spaces", { count: numRooms, numSpaces });
+                } else if (numSpaces > 0) {
+                    countsStr = _t("%(count)s rooms and 1 space", { count: numRooms, numSpaces });
+                } else {
+                    countsStr = _t("%(count)s rooms", { count: numRooms, numSpaces });
+                }
+
+                let manageButtons;
+                if (space.getMyMembership() === "join" &&
+                    space.currentState.maySendStateEvent(EventType.SpaceChild, userId)
+                ) {
+                    const selectedRelations = Array.from(selected.keys()).flatMap(parentId => {
+                        return [
+                            ...selected.get(parentId).values(),
+                        ].map(childId => [parentId, childId]) as [string, string][];
+                    });
+
+                    const selectionAllSuggested = selectedRelations.every(([parentId, childId]) => {
+                        return parentChildMap.get(parentId)?.get(childId)?.content.suggested;
+                    });
+
+                    const disabled = !selectedRelations.length || removing || saving;
+
+                    let Button: React.ComponentType<React.ComponentProps<typeof AccessibleButton>> = AccessibleButton;
+                    let props = {};
+                    if (!selectedRelations.length) {
+                        Button = AccessibleTooltipButton;
+                        props = {
+                            tooltip: _t("Select a room below first"),
+                            yOffset: -40,
+                        };
+                    }
+
+                    manageButtons = <>
+                        <Button
+                            {...props}
+                            onClick={async () => {
+                                setRemoving(true);
+                                try {
+                                    for (const [parentId, childId] of selectedRelations) {
+                                        await cli.sendStateEvent(parentId, EventType.SpaceChild, {}, childId);
+                                        parentChildMap.get(parentId).delete(childId);
+                                        if (parentChildMap.get(parentId).size > 0) {
+                                            parentChildMap.set(parentId, new Map(parentChildMap.get(parentId)));
+                                        } else {
+                                            parentChildMap.delete(parentId);
+                                        }
+                                    }
+                                } catch (e) {
+                                    setError(_t("Failed to remove some rooms. Try again later"));
+                                }
+                                setRemoving(false);
+                            }}
+                            kind="danger_outline"
+                            disabled={disabled}
+                        >
+                            { removing ? _t("Removing...") : _t("Remove") }
+                        </Button>
+                        <Button
+                            {...props}
+                            onClick={async () => {
+                                setSaving(true);
+                                try {
+                                    for (const [parentId, childId] of selectedRelations) {
+                                        const suggested = !selectionAllSuggested;
+                                        const existingContent = parentChildMap.get(parentId)?.get(childId)?.content;
+                                        if (!existingContent || existingContent.suggested === suggested) continue;
+
+                                        const content = {
+                                            ...existingContent,
+                                            suggested: !selectionAllSuggested,
+                                        };
+
+                                        await cli.sendStateEvent(parentId, EventType.SpaceChild, content, childId);
+
+                                        parentChildMap.get(parentId).get(childId).content = content;
+                                        parentChildMap.set(parentId, new Map(parentChildMap.get(parentId)));
+                                    }
+                                } catch (e) {
+                                    setError("Failed to update some suggestions. Try again later");
+                                }
+                                setSaving(false);
+                                setSelected(new Map());
+                            }}
+                            kind="primary_outline"
+                            disabled={disabled}
+                        >
+                            { saving
+                                ? _t("Saving...")
+                                : (selectionAllSuggested ? _t("Mark as not suggested") : _t("Mark as suggested"))
+                            }
+                        </Button>
+                    </>;
+                }
+
+                let results;
+                if (roomsMap.size) {
+                    const hasPermissions = space?.currentState.maySendStateEvent(EventType.SpaceChild, cli.getUserId());
+
+                    results = <>
+                        <HierarchyLevel
+                            spaceId={space.roomId}
+                            rooms={roomsMap}
+                            relations={parentChildMap}
+                            parents={new Set()}
+                            selectedMap={selected}
+                            onToggleClick={hasPermissions ? (parentId, childId) => {
+                                setError("");
+                                if (!selected.has(parentId)) {
+                                    setSelected(new Map(selected.set(parentId, new Set([childId]))));
+                                    return;
+                                }
+
+                                const parentSet = selected.get(parentId);
+                                if (!parentSet.has(childId)) {
+                                    setSelected(new Map(selected.set(parentId, new Set([...parentSet, childId]))));
+                                    return;
+                                }
+
+                                parentSet.delete(childId);
+                                setSelected(new Map(selected.set(parentId, new Set(parentSet))));
+                            } : undefined}
+                            onViewRoomClick={(roomId, autoJoin) => {
+                                showRoom(roomsMap.get(roomId), Array.from(viaMap.get(roomId) || []), autoJoin);
+                            }}
+                        />
+                        { children && <hr /> }
+                    </>;
+                } else {
+                    results = <div className="mx_SpaceRoomDirectory_noResults">
+                        <h3>{ _t("No results found") }</h3>
+                        <div>{ _t("You may want to try a different search or check for typos.") }</div>
+                    </div>;
+                }
+
+                content = <>
+                    <div className="mx_SpaceRoomDirectory_listHeader">
+                        { countsStr }
+                        <span>
+                            { additionalButtons }
+                            { manageButtons }
+                        </span>
+                    </div>
+                    { error && <div className="mx_SpaceRoomDirectory_error">
+                        { error }
+                    </div> }
+                    <AutoHideScrollbar className="mx_SpaceRoomDirectory_list" onKeyDown={onKeyDownHandler} role="tree">
+                        { results }
+                        { children }
+                    </AutoHideScrollbar>
+                </>;
+            } else {
+                content = <Spinner />;
+            }
+
+            return <>
+                <SearchBox
+                    className="mx_SpaceRoomDirectory_search mx_textinput_icon mx_textinput_search"
+                    placeholder={_t("Search names and descriptions")}
+                    onSearch={setQuery}
+                    autoFocus={true}
+                    initialValue={initialText}
+                    onKeyDown={onKeyDownHandler}
+                />
+
+                { content }
+            </>;
+        } }
+    </RovingTabIndexProvider>;
 };
 
 interface IProps {
diff --git a/src/components/views/spaces/SpacePanel.tsx b/src/components/views/spaces/SpacePanel.tsx
index 58e1db4b1d..38d530bee5 100644
--- a/src/components/views/spaces/SpacePanel.tsx
+++ b/src/components/views/spaces/SpacePanel.tsx
@@ -272,6 +272,7 @@ const SpacePanel = () => {
                     <ul
                         className={classNames("mx_SpacePanel", { collapsed: isPanelCollapsed })}
                         onKeyDown={onKeyDownHandler}
+                        role="tree"
                     >
                         <Droppable droppableId="top-level-spaces">
                             { (provided, snapshot) => (
diff --git a/src/components/views/spaces/SpaceTreeLevel.tsx b/src/components/views/spaces/SpaceTreeLevel.tsx
index 5d2f58fab2..5b32722504 100644
--- a/src/components/views/spaces/SpaceTreeLevel.tsx
+++ b/src/components/views/spaces/SpaceTreeLevel.tsx
@@ -328,7 +328,7 @@ const SpaceTreeLevel: React.FC<ITreeLevelProps> = ({
     isNested,
     parents,
 }) => {
-    return <ul className="mx_SpaceTreeLevel">
+    return <ul className="mx_SpaceTreeLevel" role="group">
         { spaces.map(s => {
             return (<SpaceItem
                 key={s.roomId}