In applications of Ricci flow, one evolves a Riemannian metric g(t) on a manifold M in order to improve its geometry. Frequently, this evolution forces changes in topology. These changes are triggered by singularity formation. The most interesting are local singularities, in which the metric remains regular on a large open subset of the manifold. In these cases, an adequate understanding of the geometry in a space-time neighborhood of the singularity enables one to perform topological-geometric surgeries. I will introduce the subject and describe a program of joint work with Sigurd Angenent (UW-Madison) in which we obtain precise asymptotic expansions for local singularities.