A differential cohomology theory (DCT) is a type of refinement of a cohomology theory (restricted to the category of smooth manifolds with corners) that contains information that is not homotopy invariant. A detailed definition of a DCT will be given, as well as axioms for a category larger than that of smooth manifolds with corners. Any two DCTs that are defined on this larger category and that refine the same cohomology theory will be naturally isomorphic.