  We consider a general class of $N\times N$ random matrices whose entries
$h_{ij}$ are independent up to a symmetry constraint, but not necessarily
identically distributed. Our main result is a local semicircle law which
improves previous results [14] both in the bulk and at the edge. The error
bounds are given in terms of the basic small parameter of the model,
$\max_{i,j} \E \abs{h_{ij}}^2$. As a consequence, we prove the universality of
the local $n$-point correlation functions in the bulk spectrum for a class of
matrices whose entries do not have comparable variances, including random band
matrices with band width $W\gg N^{1-\e_n}$ with some $\e_n>0$ and with a
negligible mean-field component. In addition, we provide a coherent and
pedagogical proof of the local semicircle law, streamlining and strengthening
previous arguments from [3,4,16].
