  We consider $N\times N$ Hermitian random matrices with independent identical
distributed entries. The matrix is normalized so that the average spacing
between consecutive eigenvalues is of order 1/N. Under suitable assumptions on
the distribution of the single matrix element, we prove that, away from the
spectral edges, the density of eigenvalues concentrates around the Wigner
semicircle law on energy scales $\eta \gg N^{-1} (\log N)^8$. Up to the
logarithmic factor, this is the smallest energy scale for which the semicircle
law may be valid. We also prove that for all eigenvalues away from the spectral
edges, the $\ell^\infty$-norm of the corresponding eigenvectors is of order
$O(N^{-1/2})$, modulo logarithmic corrections. The upper bound $O(N^{-1/2})$
implies that every eigenvector is completely delocalized, i.e., the maximum
size of the components of the eigenvector is of the same order as their average
size. In the Appendix, we include a lemma by J. Bourgain which removes one of
our assumptions on the distribution of the matrix elements.
