Classical theorems of P.A. Smith and E.E. Floyd from the 1940's and 50's concerned actions of finite p-groups on finite dimensional complexes, relating the mod p homology of the space to the mod p homology of its space of fixed points. There are now chromatic versions of these, with Morava K-theories replacing mod p homology.

I will explain the theorems that have been proved, and what still hasn't.