Derived categories of sheaves: a skimming
  These lecture notes were prepared for the workshop ``Algebraic Geometry: Presentations by Young Researchers'' in Snowbird, Utah, July 2004, and for the autumn school in Lukecin, Poland, September 2004. In six lectures I attempted to present a non-technical introduction to derived categories of sheaves on varieties, including some important results like:   -- the description of the derived category of P^n;   -- Serre duality;   -- the Bondal-Orlov result about derived categories of varieties with ample or antiample canonical class;   -- the Mukai-Bondal-Orlov-Bridgeland criterion for equivalence;   -- equivalences for K3 surfaces and abelian varieties;   -- invariance of Hochschild homology and cohomology under derived equivalence. 
Andrei Caldararu