Summary: In physics there is an urgent necessity to base some geometric models of physical phenomena on “sufficiently non-smooth” generalizations of the differentiable manifold concept. The theory of differential spaces might provide physics with such a possibility. Algebraic foundations of this theory are discussed. Differential space in the sense of Sikorski turns out to be a “geometric refinement” of the algebraic concept of ringed space, and it naturally generalizes the real manifold concept. However, it proves to be inadequate to deal with complex analytic manifolds. Mostow’s theory of differential spaces is a geometric version of the theory of structured spaces (essentially, sheaves of germs of functions on a topological space). It is shown that to naturally generalize the concept of complex analytic manifold one must suitably adapt Mostow’s concept of differential space.