  The Anderson model on the Bethe lattice is historically among the first for
which an energy regime of extended states and a separate regime of localized
states could be established. In this paper, we review recently discovered
surprises in the phase diagram. Among them is that even at weak disorder, the
regime of diffusive transport extends well beyond energies of the unperturbed
model into the Lifshitz tails. As will be explained, the mechanism for the
appearance of extended states in this non-perturbative regime are
disorder-induced resonances. We also present remaining questions concerning the
structure of the eigenfunctions and the associated spectral statistics problem
on the Bethe lattice.
