An abelian variety is a complex torus that can be embedded in projective space. A smooth complex projective variety X is of maximal Albanese dimension if it admits a morphism a:X—>A to an abelian variety A such that dim a(X)=dim X. Being of maximal Albanese dimension is a topological property and it imposes significant restrictions on the numerical invariants of the variety and on the behaviour of its linear systems. In my talk I will report on recent progress on these topics, obtained in collaboration with Miguel Angel Barja (UPC - Barcelona) and Lidia Stoppino (University’ di Pavia).