  We study the birational geometry of varieties of maximal Albanese dimension.
In particular we discuss criteria for a generically finite morphism of
varieties of maximal Albanese dimension to be birational; we give a new
characterization of Theta divisors; we study the Albanese map and refine some
of the results of Koll\'ar; finally we use these results to birationally
classify varieties with $P_3(X)=2$ and $q(X)=dim (X)$. Our method combines the
generic vanishing theorems of Green and Lazarsfeld, the theory of Fourier Mukai
transforms and the results of Koll\`ar on higher direct images of dualizing
sheaves.
