diff --git a/.circleci/config.yml b/.circleci/config.yml
@@ -141,6 +141,7 @@ commands:
                [ "$CIRCLE_PR_NUMBER" = "" ]; then
               export RELEASE_TAG='-t release'
             fi
+            mkdir -p logs
             make html O="-T $RELEASE_TAG -j4 -w /tmp/sphinxerrorswarnings.log"
             rm -r build/html/_sources
           working_directory: doc
@@ -154,20 +155,35 @@ commands:
       - run:
           name: Extract possible build errors and warnings
           command: |
-           (grep "WARNING\|ERROR" /tmp/sphinxerrorswarnings.log ||
-            echo "No errors or warnings")
+            (grep "WARNING\|ERROR" /tmp/sphinxerrorswarnings.log ||
+             echo "No errors or warnings")
+            # Save logs as an artifact, and convert from absolute paths to
+            # repository-relative paths.
+            sed "s~$PWD/~~" /tmp/sphinxerrorswarnings.log > \
+              doc/logs/sphinx-errors-warnings.log
           when: always
+      - store_artifacts:
+          path: doc/logs/sphinx-errors-warnings.log
 
   doc-show-deprecations:
     steps:
       - run:
           name: Extract possible deprecation warnings in examples and tutorials
           command: |
-           (grep DeprecationWarning -r -l doc/build/html/gallery ||
-            echo "No deprecation warnings in gallery")
-           (grep DeprecationWarning -r -l doc/build/html/tutorials ||
-            echo "No deprecation warnings in tutorials")
+            (grep -rl DeprecationWarning doc/build/html/gallery ||
+             echo "No deprecation warnings in gallery")
+            (grep -rl DeprecationWarning doc/build/html/plot_types ||
+             echo "No deprecation warnings in plot_types")
+            (grep -rl DeprecationWarning doc/build/html/tutorials ||
+             echo "No deprecation warnings in tutorials")
+            # Save deprecations that are from this absolute directory, and
+            # convert to repository-relative paths.
+            (grep -Ero --no-filename "$PWD/.+DeprecationWarning.+$" \
+             doc/build/html/{gallery,plot_types,tutorials} || echo) | \
+              sed "s~$PWD/~~" > doc/logs/sphinx-deprecations.log
           when: always
+      - store_artifacts:
+          path: doc/logs/sphinx-deprecations.log
 
   doc-bundle:
     steps:

diff --git a/.circleci/fetch_doc_logs.py b/.circleci/fetch_doc_logs.py
@@ -0,0 +1,63 @@
+"""
+Download artifacts from CircleCI for a documentation build.
+
+This is run by the :file:`.github/workflows/circleci.yml` workflow in order to
+get the warning/deprecation logs that will be posted on commits as checks. Logs
+are downloaded from the :file:`docs/logs` artifact path and placed in the
+:file:`logs` directory.
+
+Additionally, the artifact count for a build is produced as a workflow output,
+by appending to the file specified by :env:`GITHUB_OUTPUT`.
+
+If there are no logs, an "ERROR" message is printed, but this is not fatal, as
+the initial 'status' workflow runs when the build has first started, and there
+are naturally no artifacts at that point.
+
+This script should be run by passing the CircleCI build URL as its first
+argument. In the GitHub Actions workflow, this URL comes from
+``github.event.target_url``.
+"""
+import json
+import os
+from pathlib import Path
+import sys
+from urllib.parse import urlparse
+from urllib.request import urlopen
+
+
+if len(sys.argv) != 2:
+    print('USAGE: fetch_doc_results.py CircleCI-build-url')
+    sys.exit(1)
+
+target_url = urlparse(sys.argv[1])
+*_, organization, repository, build_id = target_url.path.split('/')
+print(f'Fetching artifacts from {organization}/{repository} for {build_id}')
+
+artifact_url = (
+    f'https://circleci.com/api/v2/project/gh/'
+    f'{organization}/{repository}/{build_id}/artifacts'
+)
+print(artifact_url)
+with urlopen(artifact_url) as response:
+    artifacts = json.load(response)
+artifact_count = len(artifacts['items'])
+print(f'Found {artifact_count} artifacts')
+
+with open(os.environ['GITHUB_OUTPUT'], 'w+') as fd:
+    fd.write(f'count={artifact_count}\n')
+
+logs = Path('logs')
+logs.mkdir(exist_ok=True)
+
+found = False
+for item in artifacts['items']:
+    path = item['path']
+    if path.startswith('doc/logs/'):
+        path = Path(path).name
+        print(f'Downloading {path} from {item["url"]}')
+        with urlopen(item['url']) as response:
+            (logs / path).write_bytes(response.read())
+        found = True
+
+if not found:
+    print('ERROR: Did not find any artifact logs!')
diff --git a/.github/workflows/circleci.yml b/.github/workflows/circleci.yml
@@ -1,11 +1,12 @@
 ---
+name: "CircleCI artifact handling"
 on: [status]
-permissions:
-  statuses: write
 jobs:
   circleci_artifacts_redirector_job:
-    runs-on: ubuntu-latest
     if: "${{ github.event.context == 'ci/circleci: docs-python38' }}"
+    permissions:
+      statuses: write
+    runs-on: ubuntu-latest
     name: Run CircleCI artifacts redirector
     steps:
       - name: GitHub Action step
@@ -15,7 +16,54 @@ jobs:
           artifact-path: 0/doc/build/html/index.html
           circleci-jobs: docs-python38
           job-title: View the built docs
-      - name: Check the URL
-        if: github.event.status != 'pending'
+
+  post_warnings_as_review:
+    if: "${{ github.event.context == 'ci/circleci: docs-python38' }}"
+    permissions:
+      contents: read
+      checks: write
+      pull-requests: write
+    runs-on: ubuntu-latest
+    name: Post warnings/errors as review
+    steps:
+      - uses: actions/checkout@v3
+
+      - name: Fetch result artifacts
+        id: fetch-artifacts
+        run: |
+          python .circleci/fetch_doc_logs.py "${{ github.event.target_url }}"
+
+      - name: Set up reviewdog
+        if: "${{ steps.fetch-artifacts.outputs.count != 0 }}"
+        uses: reviewdog/action-setup@v1
+        with:
+          reviewdog_version: latest
+
+      - name: Post review
+        if: "${{ steps.fetch-artifacts.outputs.count != 0 }}"
+        env:
+          REVIEWDOG_GITHUB_API_TOKEN: ${{ secrets.GITHUB_TOKEN }}
+          REVIEWDOG_SKIP_DOGHOUSE: "true"
+          CI_COMMIT: ${{ github.event.sha }}
+          CI_REPO_OWNER: ${{ github.event.repository.owner.login }}
+          CI_REPO_NAME: ${{ github.event.repository.name }}
         run: |
-          curl --fail ${{ steps.step1.outputs.url }} | grep $GITHUB_SHA
+          # The 'status' event does not contain information in the way that
+          # reviewdog expects, so we unset those so it reads from the
+          # environment variables we set above.
+          unset GITHUB_ACTIONS GITHUB_EVENT_PATH
+          cat logs/sphinx-errors-warnings.log | \
+            reviewdog \
+              -efm '%f\:%l: %tEBUG: %m' \
+              -efm '%f\:%l: %tNFO: %m' \
+              -efm '%f\:%l: %tARNING: %m' \
+              -efm '%f\:%l: %tRROR: %m' \
+              -efm '%f\:%l: %tEVERE: %m' \
+              -efm '%f\:%s: %tARNING: %m' \
+              -efm '%f\:%s: %tRROR: %m' \
+              -name=sphinx -tee -fail-on-error=false \
+              -reporter=github-check -filter-mode=nofilter
+          cat logs/sphinx-deprecations.log | \
+            reviewdog \
+              -efm '%f\:%l: %m' \
+              -name=examples -tee -reporter=github-check -filter-mode=nofilter