YouTube Music has added an incredibly useful feature today, one that amazingly hasn't been in all this time. We're talking about proper support for saving stuff offline.
Up until this point, YouTube Music's only option in this realm was an auto-generated "offline mixtape", which was created "based on your previous listening history". In other words, you had no control over what went into it.
From now on, though, if you're a YouTube Red subscriber, YouTube music finally lets you choose what to save for offline listening or viewing - you can pick any song, album, or playlist to download. You also get to select whether you want to save just the audio, SD video, or HD video (when possible). Once you've added things to your offline library, to access it you tap your profile photo and select Offline.