In this talk we will first introduce the boundary rigidity problem: "When is a Riemannian metric determined by the distance between boundary points?". Our goal is to give an outline of the proof of the recent theorem of Pestov and Uhlmann that this is true for "simple" two dimensional manifolds. Here by "simple" we mean a metric on the disk where every geodesic is the unique minimizing path between its endpoints and the boundary is strictly convex. This is the first such result without any curvature assumption. The proof uses a number of other results that we will discuss. The main new ingredient is that knowing the distance between boundary points is enough to determine the "Dirichlet to Neumann map" DN. The domain and range of DN consists of functions defined on the boundary. DN(f) is the normal derivative of the harmonic function u whose boundary values are f.