A family of hypersurfaces $M_t\subset R^{n+1}$ evolves by mean curvature flow (MCF) if the velocity at each point is given by the mean curvature vector. MCF can be viewed as a geometric heat equation, deforming surfaces towards optimal ones. If the initial surface M_0 is convex, then the evolving surfaces M_t become rounder and rounder and converge (after rescaling) to the standard sphere S^n. The central task in studying MCF for more general initial surfaces is to analyze the formation of singularities. For example, if M_0 looks like a a dumbbell, then the neck will pinch off preventing one from continuing the flow in a smooth way. To resolve this issue, one can either try to continue the flow as a generalized weak solution or try to perform surgery (i.e. cut along necks and replace them by caps). These ideas have been implemented in the last 15 years in the deep work of White and Huisken-Sinestrari, and recently Kleiner and I found a streamlined and unified approach (arXiv: 1304.0926, 1404.2332). In this lecture, I will survey these developments for a general audience.