  In this article we survey recent results on rigid dualizing complexes over
commutative algebras. We begin by recalling what are dualizing complexes. Next
we define rigid complexes, and explain their functorial properties. Due to the
possible presence of torsion, we must use differential graded algebras in the
constructions. We then discuss rigid dualizing complexes. Finally we show how
rigid complexes can be used to understand Cohen-Macaulay homomorphisms and
relative dualizing sheaves.
