  Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the
distribution of the $(i,j)$ matrix element is given by a probability measure
$\nu_{ij}$ with a subexponential decay.
  Let $\sigma_{ij}^2$ be the variance for the probability measure $\nu_{ij}$
with the normalization property that $\sum_{i} \sigma^2_{ij} = 1$ for all $j$.
Under essentially the only condition that $c\le N \sigma_{ij}^2 \le c^{-1}$ for
some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue
spacing statistics of $H$ in the bulk of the spectrum coincide with those of
the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for
band matrices with bandwidth $M$ the local semicircle law holds to the energy
scale $M^{-1}$.
