D-modules and local cohomology

The concept of modules over a ring with differential operators is an old one, but its relevance to commutative algebra dates from 1993, when Lyubeznik discovered that the local cohomology module of a polynomial ring with supports in any ideal is a finitely generated D-module, even though it is rarely finitely generated over the polynomial ring. In fact these modules are what is called holonomic, i.e. of finite length over the ring of differential operators. I will explain the ideas behind this discovery, and tell how it has helped to prove new finiteness theorems about these mysterious modules.