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: http://www.columbia.edu/~dl3133/MFGSpring2018.pdf Background material on mean-field interacting processes by Golse: https://arxiv.org/abs/1301.5494 Books: 1). There is a two volume ...

Only top voted, non community-wiki answers of a minimum length are eligible