  We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The
matrix is normalized so that the average spacing between consecutive
eigenvalues is of order $1/N$. We study the connection between eigenvalue
statistics on microscopic energy scales $\eta\ll1$ and (de)localization
properties of the eigenvectors. Under suitable assumptions on the distribution
of the single matrix elements, we first give an upper bound on the density of
states on short energy scales of order $\eta \sim\log N/N$. We then prove that
the density of states concentrates around the Wigner semicircle law on energy
scales $\eta\gg N^{-2/3}$. We show that most eigenvectors are fully delocalized
in the sense that their $\ell^p$-norms are comparable with
$N^{{1}/{p}-{1}/{2}}$ for $p\ge2$, and we obtain the weaker bound
$N^{{2}/{3}({1}/{p}-{1}/{2})}$ for all eigenvectors whose eigenvalues are
separated away from the spectral edges. We also prove that, with a probability
very close to one, no eigenvector can be localized. Finally, we give an optimal
bound on the second moment of the Green function.
