  We consider $N\times N$ symmetric or hermitian random matrices with
independent, identically distributed entries where the probability distribution
for each matrix element is given by a measure $\nu$ with a subexponential
decay. We prove that the local eigenvalue statistics in the bulk of the
spectrum for these matrices coincide with those of the Gaussian Orthogonal
Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE), 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. We also show that
the Wigner semicircle law holds locally on the smallest possible scales and we
prove that eigenvectors are fully delocalized and eigenvalues repel each other
on arbitrarily small scales.
