In this paper we present a new notion of curvature for cell complexes. For each p, we define a pth combinatorial curvature function, which assigns a number to each p-cell of the complex. The curvature of a p-cell depends only on the relationships between the cell and its neighbors. In the case that p = 1, the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds. We begin by deriving a combinatorial analogue of Bochner’s theorems, which demonstrate that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing the properties of the combinatorial Ricci curvature with those of its Riemannian avatar.