Real toric manifolds are the real loci of nonsingular complete toric varieties. In this paper, we calculate the integral cohomology groups of real toric manifolds in terms of the combinatorial data contained in the underlying simplicial fans, generalizing a formula due to Cai and Choi. We also present a combinatorial formula to describe the ring structure, modulo the ideal consisting of elements of order $2$. As an application, a combinatorial-algebraic criterion is established for when a real toric manifold is spin$^c$.