We outline the construction of the derived category D(M) of an abelian category M. We then define left and right derived functors. We introduce K-projective, K-injective and K-flat resolutions, and prove existence of some derived functors when M is either the category of modules over a ring, or the category of sheaves of modules over a sheaf of rings. DG algebras and their derived categories will also be mentioned. Next we discuss some more specialized topics: dualizing complexes (commutative and noncommutative), two-sided tilting complexes, derived Morita theory (for DG algebras), and perverse sheaves.