Pugh (1967) showed a C^1-generic diffeomorphism of a compact manifold has a dense set of periodic points. Pugh-Robinson (1983) proved several extensions to other classes of maps, including to volume-preserving diffeomorphisms. The extension to higher regularity (C^r-generic instead of C^1, for r >= 2) has remained largely open since then. I will discuss some joint work with Cristofaro-Gardiner and Zhang showing that a C^\infty-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points.