  We study the universality of spectral statistics of large random matrices. We
consider $N\times N$ symmetric, hermitian or quaternion self-dual random
matrices with independent, identically distributed entries (Wigner matrices)
where the probability distribution for each matrix element is given by a
measure $\nu$ with a subexponential decay. Our main result is that the
correlation functions of the local eigenvalue statistics in the bulk of the
spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the
Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE),
respectively, in the limit $N\to \infty$. Our approach is based on the study of
the Dyson Brownian motion via a related new dynamics, the local relaxation
flow. As a main input, we establish that the density of eigenvalues converges
to the Wigner semicircle law and this holds even down to the smallest possible
scale, and, moreover, we show that eigenvectors are fully delocalized. These
results hold even without the condition that the matrix elements are
identically distributed, only independence is used. In fact, we give strong
estimates on the matrix elements of the Green function as well that imply that
the local statistics of any two ensembles in the bulk are identical if the
first four moments of the matrix elements match. Universality at the spectral
edges requires matching only two moments. We also prove a Wegner type estimate
and that the eigenvalues repel each other on arbitrarily small scales.
