Is it legal to embed full movies from YouTube on a website?

So there are a lot of full length movies on YouTube. This shocked me at first considering YouTube is doing nothing about this.

I started thinking about things to do with this... So, would it be illegal to create an aggregator of sorts that curates all of the full length movies on YouTube and displays them on a website using YouTube's embed feature? I wouldn't be hosting the movies (that is YouTube's error I suppose).

Any thoughts on this would be appreciated.

*As a sidenote, here's a very rough mockup of a website for this concept. Lots of the buttons don't work, but I'd still love to hear some thoughts on it :)

http://www.wix.com/dkoch15/movienites