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The general formulation of the universality conjecture for disordered systems states that there are two distinctive regimes depending on the energy and the disorder strength. In the strong disorder regime, the eigenvectors are localized and the local spectral statistics are Poisson. In the weak disorder regime, the eigenvectors are delocalized and the local statistics coincide with those of a Gaussian matrix ensemble. Random band matrices are natural intermediate models to study the eigenvalue statistics and quantum propagation in disordered systems, as they interpolate between mean-field-type Wigner matrices and random Schrödinger operators. In particular, band matrices provide a means of probing the localization-delocalization transition. I shall report on recent joint work with L. Erdos. We consider a large class of random band matrices H with band width W, and prove that the quantum time evolution generated by H is diffusive up to time scales of order W^{d/3}, where d is the number of spatial dimensions. We also derive an explicit formula for the diffusion constant. As a corollary, we prove that the localization length of an arbitrarily large majority of eigenvectors is larger than a factor W^{d/6} times the band width W.