Exact forms are closed, so the vector spaces of k-forms along with the exterior derivative are a cochain complex. The vector spaces of closed forms modulo exact forms are called the de Rham cohomology groups. The wedge product endows the direct sum of these groups with a ring structure.
De Rham's theorem, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold M, these groups are isomorphic as real vector spaces with the singular cohomology groups Hp(M;R). Further, the two cohomology rings are isomorphic (as graded rings).