Downloading an entire Youtube playlist is possible. You may be interested in downloading entire playlists for offline viewing later to study for an exam or just to watch music videos.
For android you can use TubeMate. Make sure you don’t download TubeMate from Google Playstore because there are a lot of malware infected versions out there. Visit the below site for links to trusted versions.
With Tubemate I was able to save all videos one by one automatically to my Samsung Note 5. I just chose the Playlist and tapped the download icon and it took care of the rest.
If you are trying to download entire playlists or channels on PC, there are PC programs that accomplish this, but be careful because Malware is very common. A lot of Malware makers pay the people that make useful PC programs–so just make sure you download from a trusted source and avoid agreeing to or check marking anything that asks you to install something or change your default browser search engine.
The previous extensions for Chrome and Mozilla Firefox no longer work, but someone may create a new extension that allows people to download Youtube channels and playlists.
If you are looking to save Youtube videos to your iPhone or iPad–ProTube is the tool for you. I know you can save videos, but can’t guarantee if you can save entire playlists with this app.