  Generalized tensor analysis in the sense of Colombeau's construction is
employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In
particular, after deriving several characterizations of invertibility in the
algebra of generalized functions we define the notions of generalized
pseudo-Riemannian metric, generalized connection and generalized curvature
tensor. We prove a ``Fundamental Lemma of (pseudo-)Riemannian geometry'' in
this setting and define the notion of geodesics of a generalized metric.
Finally, we present applications of the resulting theory to general relativity.
