These notes are based on the Fermi Lectures delivered at the Scuola Normale Superiore, Pisa, in June 2001. The principal aim of the lectures was to present the structure theory developed by Toby Colding and myself, for metric spaces which are Gromov-Hausdorff limits of sequences of Riemannian manifolds which satisfy a uniform lower bound of Ricci curvature. The emphasis in the lectures was on the non-collapsing situation. A particularly interesting case is that in which the manifolds in question are Einstein (or Kahler-Einstein). Thus, the theory provides information on the manner in which Einstein metrics can degenerate.