The moduli space of (pointed) Riemann surfaces can be naturally compactified by adding various degenerate surfaces. The resulting Delign-Mumford-Knudsen compactification exhibits many interesting geometric phenomena, such as pinching geodesics and bubbling off of spheres. I will discuss these phenomena and recent results (joint with Richard Melrose) on the behavior of the constant curvature metric when approaching the boundary of the compactification. This involves analyzing the Laplace operator on a family of degenerating surfaces. I will also talk about its application to the study of the asymptotics of the Weil-Petersson metric on the moduli space.