On a compact Kahler manifold, the Kahler-Ricci flow is a differential equation whose solution, when it exists, is a family of Kahler metrics on the manifold. Initially introduced to produce canonical metrics on complex manifolds, the Kahler-Ricci flow is now a major tool in Kahler geometry. I aim to give a brief account of how the Kahler-Ricci flow arose in the study of canonical metrics, and to explain, in the case of complex surfaces, how the Kahler-Ricci behaves as an analytic version of the minimal model program. Speaker(s): Matt Stevenson (UM)