We consider the inverse problem of travel time tomography: Can one determine the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium? This problem arises in global seismology, medical imaging, oceanography and many others.  This inverse problem can be recast as a geometric problem: can one determine the Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function or the scattering relation?