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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.
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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 ...
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