The special and general theories of relativity are briefly reviewed and the conceptual basis for a nonlocal generalization of relativity theory is presented. Nonlocality - in the sense of an influence ("memory") from the past that endures - could be a natural feature of the universal gravitational interaction. Nonlocal gravity is formally analogous to the nonlocal electrodynamics of media. The nonlocal aspect of gravity can simulate dark matter. The implications of nonlocal gravity theory for the problem of dark matter in galaxies, clusters of galaxies, and cosmology are mentioned.