The lecture notes on mean-field games of Lions, from a course at the Collège de France, have been typed out by Pierre Cardaliaguet. They address both optimal control and dynamic programming. I would think it will be hard to beat these for an authoritative exposition. For a summary of open problems you could look here.
Here's my go to links:
PDE flavor notes by Ryzhik: https://math.stanford.edu/~ryzhik/STANFORD/MEAN-FIELD-GAMES/notes-mean-field.pdf
Probability flavor notes by Lacker:
Background material on mean-field interacting processes by Golse:
1). There is a two volume ...