  We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$
dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset
\Z^d$, are independent, uniformly distributed random variables if $\abs{x-y}$
is less than the band width $W$, and zero otherwise. We prove that the time
evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on
time scales $t\ll W^{d/3}$. We also show that the localization length of an
arbitrarily large majority of the eigenvectors is larger than a factor
$W^{d/6}$ times the band width. All results are uniform in the size
$\abs{\Lambda}$ of the matrix.
