Just figured it out today! I listen to music in the YouTube app, and then lock my screen or close the app, only to realize the music stops. For years, I've thought it not possible to run music in the background from YouTube. But I'm happy to say that I was wrong.
How to do it:
1. Open the YouTube app and play a video.
2. Go to the home screen, the music will stop playing.
3. Open the multitask bar, and scroll to the left, where the music controls are. The YouTube app should be depicted on the right.
4. Just hit play, and go on with your business.
Did you know of this? If so, why didn't you tell me!? :P