Sharp cutoff for walks on graphs and groups

In the `80 Diaconis and others have observed that some naturally occurring Markov chains exhibit a cutoff phenomenon: the distance from the stationary distribution drops from almost maximal to almost zero over a short period of time. It is conjectured that all transitive expander graphs exhibit this phenomenon, but so far very few examples are known. I will survey and explain some results on cutoff for walks on Ramanujan graphs and complexes, and their relation to expansion in groups.