  We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d
\geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \in
(\bZ/L\bZ)^d$, are independent, centred random variables with variances $s_{xy}
= \E |h_{xy}|^2$. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the
band width $W$. In one dimension we prove that the eigenvectors of $H$ are
delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix
entries $\abs{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging
and we compute $\E \abs{G_{xy}}^2$. We show that, as $L\to\infty$ and $W\gg
L^{4/5}$, the behaviour of $\E |G_{xy}|^2$ is governed by a diffusion operator
whose diffusion constant we compute. Similar results are obtained in higher
dimensions.
