This book contains a written version of lectures presented at the meeting of the Société Mathématique de France held in the summer of 2004 at Dijon. It consists of four contributions. The first one (by S. Crovisier) is devoted to a description of the space of orbits (and in particular the space of periodic orbits) of a chosen volume preserving diffeomorphism f of a smooth compact connected surface with a given smooth volume. To avoid pathological cases, the author considers C1-generic surface diffeomorphisms. The next contribution (by J. Franks) studies distortion elements in groups of surface diffeomorphisms (and, for comparison, also for circle diffeomorphisms). The results are applied to group actions on surfaces preserving a Borel measure. Three different topics and their relations are treated in the third contribution (by J.-M. Gambaudo): topological invariants associated with flows on three-dimensional manifolds, the space of configurations of an incompressible fluid on an oriented manifold, and a structure of the group of diffeomorphisms (preserving a given area form and isotopic to the identity) of a compact oriented surface. Finally, P. Le Calvez studies in the last contribution various versions of the Brouwer plane translation theorem and its relation to the study of homeomorphisms of surfaces (including a proof of the Conley conjecture in the case of a compact surface of genus greater than zero).