Two basic tools provided by the theory of coherent sheaves on projective varieties are vanishing of the cohomology and global gen- eration. On irregular varieties there are wonderful natural weakenings of such notions: generic vanishing and continuous global generation. They go together with the systematic use of the Fourier-Mukai trans- form. These ideas have their roots in three important bodies of work:

• Mukai's theory of the Fourier-Mukai trasform;
• Green-Lazarsfeld's generic vanishing theorems, and their applica- tions to the geometry of irregular varieties due to Ein, Lazarsfeld, Hacon and others;
• Kempf's work on theta-functions. In my lectures I will focus on such concepts. Moreover I will describe several concrete applications to geometric and algebraic problems, as syzygies of abelian varieties, special subvarieties of abelian varieties, invariants and pluricanonical maps of irregular varieties of maximal Albanese dimension.