Abstract: The Ricci flow is a geometric flow on Riemannian manifolds which has recently produced a number of exciting breakthroughs in geometric analysis, topology, and Riemannian geometry. Gradient Ricci solitons appear prominently in some of these developments as geometric fixed points of the flow and as singularity models. On the other hand, Ricci solitons also have a natural interpretation in terms of elliptic problems on Riemannian manifolds with measure. In this talk we will explore this perspective and survey some recent classification results that have arisen from this viewpoint. A natural connection between gradient Ricci solitons and warped product Einstein metrics also arises and we will discuss some recent developments in understanding this connection.