Abstract: In applications of Ricci flow, one evolves a Riemannian metric g(t) on a manifold M to improve its geometry. This evolution often forces changes in topology, changes that are triggered by singularity formation. The most interesting are local singularities, in which the metric remains regular on an 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 aspects of a program with Sigurd Angenent in which we obtain precise asymptotic expansions for local singularities. The talk will be suitable for a general audience.