This series of talks is an introduction to the boundary rigidity problem. A long term goal would be to determine a Riemannian metric on a manifold with boundary from the distances between its boundary points. This would have applications in areas from medical imaging to

seismology. Unfortunately, it is not always possible to do this. The boundary rigidity problem is to determining when it is possible. We

consider Riemannian manifolds (M,B,g) with boundary B and metric g. We let d, the "boundary distance function", be the real valued function on BxB giving the distance in M (i.e. the "chordal distance") between boundary points. The question is whether there is a unique g for a given d (up to an isometry which leaves the boundary fixed). We will talk about the various conjectures, theorems and counter examples that have been developed over the years.