Abstract: The fundamental solution to the heat equation on Euclidean space has the special property that its graph at any positive time is related to its graph at any other positive time by a scale factor and a rescaling of the spatial variable. This phenomenon is a kind of self-similarity: modulo a change of variables and a rescaling, the geometric features of the graph are invariant in time. For the Ricci flow, the analogous self-similar solutions are known as Ricci solitons: generalized fixed points of the equation which evolve only by rescaling and pull-back by diffeomorphisms of the underlying manifold. Though their evolutions are trivial from a dynamical and geometric point of view, these solutions turn out to play a central role in the analysis of the long-time behavior of the Ricci flow. They are also of independent interest as generalizations of Einstein manifolds. In this talk, we will focus on the class of noncompact shrinking Ricci solitons, which model the geometry of solutions to the Ricci flow near developing finite-time singularities. The class of such solutions exhibits a high degree of rigidity in low dimensions, and all known examples are asymptotic either to cones or to products near infinity. We will discuss some fundamental examples and survey some classification results which exhibit the close connection between a complete noncompact shrinker and its geometry at infinity. We will also discuss some current open questions with applications to the Ricci flow in dimensions four and higher.