In this Research Note, a generalization of the subject of differential geometry is developed, using techniques and results of nonstandard analysis. This generalization is found to correspond to approximations of classical manifolds by set-theoretic near manifold structures. Schlesinger develops several applications of the theory in the fields of topological dynamical systems, the question of stability of geodesic incompleteness (which is relevant to the problem of singularities in general relativity) and the deformation theory of manifolds. In the latter case, deformations induced by first cohomology can be introduced without encountering the restriction to compact manifolds as in the case of classical Kodaira-Spencer theory. This new deformation theory is then applied to a problem in twistor theory, thereby achieving a generalization of the nonlinear graviton construction of Penrose.