The dynamics of continuous interval maps admit a remarkably rich topological description, including the structure of attractors, criteria for positive entropy, and the density of periodic points in the recurrent set. In recent years, several extensions of these results have been obtained for dissipative diffeomorphisms of the disc, including Hénon maps. In this talk, I will survey some of these developments and present a new closing lemma, proved in collaboration with Enrique Pujals.