YouTube has allowed offline downloads of videos for several years now, but only if you were lucky enough to live in a country where it's supported. (I know India at least had the function in both the full YouTube app and in YouTube Go.) Now the option seems to be rolling out to more countries. One tipster in Iraq told us about it and I was able to confirm it on my device and several of my friends in Lebanon. Other AP team members in the US, UK, and Japan don't have it, so this appears to still be a geo-limited function with what could be a Levant-only rollout, a Middle Eastern one, or a wider one for more countries. We rely on you, dear readers, to let us know where it's working and where it isn't.
If you have offline downloads, you'll see the blue banner above pointing you to the button when you open your first video. Tapping on Download pops up a selection menu with different resolutions and the full size of the video in each one as well as a checkbox to remember your setting. Once done, the download starts and you'll see it in your notification bar.
Download preferences and notification.
You can follow the progress of the video's download in the notifications area or in the new Downloads section in the Library tab. There you can also pause a download while it's happening or delete it from your device.
Download progress and finished.
You can play downloaded videos any time, with or without an internet connection, and they won't eat up your data consumption. It's worth noting though that the Download button is disabled for music videos
Downloaded videos playback (left) and music videos can't be downloaded (right).
Do let us know if you see the option on your device now, if it's new or if you've had it for a while, and where you live so we can form a better understanding of Google's rollout.