We discuss a relation between the structure of derived categories of smooth
projective varieties and their birational properties. We suggest a possible
definition of a birational invariant, the derived category analogue of the
intermediate Jacobian, and discuss its possible applications to the geometry of
prime Fano threefolds and cubic fourfolds.Comment: Lecture notes for the CIME-CIRM summer school, Levico Terme, June
  22--27, 2015; 26 page