Riemannian metrics are the simplest generalizations of Euclidean geometry to smooth manifolds. The Ricci curvature of a metric measures, in an averaged sense, how the geometry deviates from being flat. The tensor −2Ric can be viewed as a Laplacian acting on the metric, so Hamilton’s Ricci flow ∂tg=−2Ric is, morally, the heat equation for metrics. In this expository talk, based on the work of others, we introduce the Ricci flow through visual depictions of how singularities may form and discuss qualitative aspects of the geometry near singularities.