The subject of Hamilton’s Ricci flow lies in the more general field of geometric flows, which in turn lies in the even more general field of geometric analysis. Ricci flow deforms Riemannian metrics on manifolds by their Ricci tensor, an equation which turns out to exhibit many similarities with the heat equation. Other geometric flows, such as the mean curvature flow of submanifolds demonstrate similar smoothing properties. The aim for many geometric flows is to produce canonical geometric structures by deforming rather general initial data to these structures. Depending on the initial data, the solutions to geometric flows may encounter singularities where at some time the solution can no longer be defined smoothly. For various reasons, in Ricci flow the study of the qualitative aspects of solutions, especially ones which form singularities, is at present more amenable in dimension 3. This is precisely the dimension in which the Poincar ́e Conjecture was originally stated; the higher dimensional generalizations have been solved by Smale in dimensions at least 5 and by Freedman in dimension 4. Remaining in dimension 3, a vast generalization of this conjecture was proposed by Thurston, called the Geometrization Conjecture, which roughly speaking, says that each closed 3-manifold admits a geometric decomposition, i.e., can be decomposed into pieces which admit complete locally homogeneous metrics.