I am curious to know what the current research of random matrix theory is focussing on. The moment method has been long considered not very practical due to all moments assumption and the highly cumbersome calculations associated with it. That's where analytic tools like orthogonal polynomials, stieltjes transforms,etc. come into play.
The symmetric matrix case has been long settled. Then Tao and Vu established universality of circular law for iid complex matrices. There is a large theory of CLT for linear spectral statistics.
So it seems to be a natural question to ask: what properties of random matrices are people trying to understand now?
EDIT: It seems that some research is now going into re-proving the existing results using recurrence relations from Physics. What are some matrix models that people want to understand now? Models that nobody knows how to proceed in?