Abstract
Large random matrices tend to exhibit universal spectral fluctuations. Besides overviewing the well-known Wigner-Dyson and Tracy-Widom universality for Hermitian Wigner matrices, we present new analogous results for non-Hermitian matrices. In particular, we establish edge universality, CLT for linear statistics and a precise three-term asymptotic expansion for the rightmost eigenvalue of an n by n random matrix with independent identically distributed complex entries as n tends to infinity.
