We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional,
orientable manifold with respect to a natural metric called the L2 metric. The primary motivation for studying this problem comes from Teichm¨uller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichm¨uller space with respect to a class of metrics
that generalize the Weil-Petersson metric.