  Let $f: X\to Y$ be a surjective morphism of smooth $n$-dimensional projective
varieties, with $Y$ of maximal Albanese dimension. Hacon and Pardini studied
the structure of $f$ assuming $P_m(X)=P_m(Y)$ for some $m\geq 2$. We extend
their result by showing that, under the above assumtions, $f$ is birationally
equivalent to a quotient by a finite abelian group. We also study the
pluricanonical map of varieties of maximal Albanese dimesnion. The main result
is that the linear series $|5K_X|$ incuces the Iitaka model of $X$.
