Google Assistant's ability to manage lists has been... less than ideal. When Assistant first went live in 2016, it could add items to a shopping list, which then appeared in Google Keep. That was a nice start, but then Google decided to move the shopping list to the Home app with integration for Google Express (now Google Shopping). At long last, Assistant is finally receiving first-class support for notes and lists... maybe. Fingers crossed.
Google first announced in mid-2018 that integration with Google Keep, Any.do, and other services was in the works for Assistant. Then a Google employee reaffirmed in October 2018 that the feature was still in development, and a blog post from the following month just said "soon." Google again said to-dos and list management was coming in a January blog post, but that section was later deleted. That makes this the fifth time Google has announced this functionality.
Google didn't limit the new list/note management feature to its own Keep app, either — it also works with Any.do, AnyList, and Bring. Just like with music and video services, you can add accounts and choose your default from the Google Assistant settings (Assistant settings > Services > Notes & Lists). Interestingly, support for Todoist was mentioned in previous announcements, but is now missing.
Here's hoping more services are added in the future, like Evernote and OneNote. This functionality should start rolling out today, so it might be a little while before you see it on your own devices.
Google may actually keep its promise this time. Some users have started seeing the new Notes & Lists menu under Assistant settings > Services. It's a server-side rollout, though, so if you have it, you can set up synchronization with Keep or any of the other services. If you don't, you can join me in impatiently waiting for it. Thanks, Zachary Kew-Denniss!
Live for all
The Notes & Lists menu in Assistant's settings appears to be live for everyone now — or at least it is for all the accounts and devices we checked. You can pick the service you want to integrate with Assistant there. If you'd like to use this for a shopping list, you should start by renaming the one Assistant automatically created for you under Settings > Services > Shopping list to something other than "shopping," then create a Shopping list in Keep, Any.do, Bring, or AnyList. With that done, Assistant should add to the one you're using, and not the legacy list.
Another quirk we've discovered is that Google Keep is recognized as a list provider only. So all notes you have in the app that don't have checkmarks (i.e. text or audio or image notes, not lists) won't be seen by Assistant. You can't ask it to show your notes, though you seem to be able to create new text ones.
For more nuances on how to set this integration as well as all the voice commands you can say, check Google's support pages for using notes on your phone or speaker, and using lists on your phone or speaker.