Abstract. I will describe the theory of Hénon attractors and Hénon-like attractors with chaotic behaviour and the corresponding ergodic theory: existence of SRB measures and the description of their basins, decay of correlation, stability under random perturbations. The maps in Hénon- and Hénon-like case are diffeomorphims. Recently, Pesin and Yurchenko have noticed similar phenomena for non-inversible maps appearing from coupled map lattice models of certain PDES. I will discuss an attempt to generalize the theory of Hénon- and Hénon-like maps to certain non-invertible cases (work in progress).