  Two short seminal papers of Margulis used Kazhdan's property $(T)$ to give,
on the one hand, explicit constructions of expander graphs, and to prove, on
the other hand, the uniqueness of some invariant means on compact simple Lie
groups. These papers opened a rich line of research on expansion and spectral
gap phenomena in finite and compact simple groups. In this paper we survey the
history of this area and point out a number of problems which are still open.
