  Derived categories were invented by Grothendieck and Verdier around 1960, not
very long after the "old" homological algebra (of derived functors between
abelian categories) was established. This "new" homological algebra, of derived
categories and derived functors between them, provides a significantly richer
and more flexible machinery than the "old" homological algebra. For instance,
the important concepts of dualizing complex and tilting complex do not exist in
the "old" homological algebra.
  This paper is an edited version of the notes for a two-lecture minicourse
given at MSRI in January 2013. Sections 1-5 are about the general theory of
derived categories, and the material is taken from my manuscript "A Course on
Derived Categories" (available online). Sections 6-9 are on more specialized
topics, leaning towards noncommutative algebraic geometry.
