  Eugene Wigner's revolutionary vision predicted that the energy levels of
large complex quantum systems exhibit a universal behavior: the statistics of
energy gaps depend only on the basic symmetry type of the model.
  Simplified models of Wigner's thesis have recently become mathematically
accessible. For mean field models represented by large random matrices with
independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture
asserts that the local eigenvalue statistics are universal. For invariant
matrix models, the eigenvalue distributions are given by a log-gas with
potential $V$ and inverse temperature $\beta = 1, 2, 4$. corresponding to the
orthogonal, unitary and symplectic ensembles. For $\beta \not \in \{1, 2, 4\}$,
there is no natural random matrix ensemble behind this model, but the analogue
of the WDGM conjecture asserts that the local statistics are independent of
$V$.
  In these lecture notes we review the recent solution to these conjectures for
both invariant and non-invariant ensembles. We will discuss two different
notions of universality in the sense of (i) local correlation functions and
(ii) gap distributions.
  We will demonstrate that the local ergodicity of the Dyson Brownian motion is
the intrinsic mechanism behind the universality. In particular, we review the
solution of Dyson's conjecture on the local relaxation time of the Dyson
Brownian motion. Additionally, the gap distribution requires a De
Giorgi-Nash-Moser type H\"older regularity analysis for a discrete parabolic
equation with random coefficients.
  Related questions such as the local version of Wigner's semicircle law and
delocalization of eigenvectors will also be discussed. We will also explain how
these results can be extended beyond the mean field models, especially to
random band matrices.
