As discussed in Anurag's lectures, the ring of differential operators over a polynomial ring has many nice homological and representation-theoretic properties; these yield striking applications to finiteness properties of local cohomology modules. More generally, one can associate to any K-algebra a ring of differential operators. Outside of the polynomial case, rings of differential operators are much less well understood, and most of the results from the polynomial case fail in general. In this talk we will survey some results, both positive and negative, on the extent to which differential operators over singular rings behave as in the regular case.