  We survey recent mathematical results about the spectrum of random band
matrices. We start by exposing the Erd{\H o}s-Schlein-Yau dynamic approach, its
application to Wigner matrices, and extension to other mean-field models. We
then introduce random band matrices and the problem of their Anderson
transition. We finally describe a method to obtain delocalization and
universality in some sparse regimes, highlighting the role of quantum unique
ergodicity.
