A problem formulated by I.M. Gelfand in the 1950s is to reconstruct the metric tensor of a compact Riemannian manifold with boundary, from data on the spectrum of its Laplace operator, with the Neumann boundary condition, and the behavior at the boundary of the normalized eigenfunctions. \vskip .1in \noindent The first ingredient that goes into the resolution of such an ``inverse problem'' is a uniqueness theorem, but further work beyond establishing uniqueness is required. This arises because of the ``ill posedness'' associated with inverse problems. That is, various ``large'' perturbations of the unknown region can yield small perturbations of the observed data. The key to stabilizing an ill-posed inverse problem is to have appropriate a priori knowledge of the unknown domain so that a search for the solution can be confined to a ``compact'' family of possible domains. In this context, the suitable notion is that of Gromov compactness, and one key to stabilizing Gelfand's inverse problem involves establishing such compactness. This is done under fairly weak hypotheses on the geometry of the unknown domain, including bounds on its curvature (to be precise, its Ricci tensor) and on the curvature of its boundary. Estimates for solutions to a naturally occuring elliptic boundary value problem for the metric tensor play a central role. \vskip .1in \noindent The speaker will discuss some of these matters, which have been treated in joint work with M. Anderson, A. Katsuda, Y. Kurylev, and M. Lassas.