We will survey some recent results, open questions, and problems related to the following "Boundary Rigidity" question: Let g0 be a Riemannian metric on a compact manifold M with boundary. Then g0 induces the boundary distance function, d0(p,q), that represents the distance in the g0 metric between the boundary points p and q (i.e. the length of the shortest path in M between p and q). We will consider the question of to what extent d0 determines g0. In particular g0 is called "boundary rigid" if every other metric g1 on M with the same boundary distance function (i.e. d1=d0) must be isometric to g0. There are examples of manifolds that are not boundary rigid and there are some that are known to be boundary rigid. This type of problem is related to problems in seismology and medical imaging as well as to conjugacy rigidity of geodesic flows, spectral rigidity for compact manifolds without boundary, and rigidity questions concerning the Dirichlet to Neumann map.