Kirillov-Reshetikhin (KR) modules form an interesting class of finite-dimensional modules of quantum affine algebras. Although a finite-dimensional irreducible module of quantum affine algebras does not have a crystal basis in general, it is believed that a KR module has a crystal basis. Polytopes are common in the study of KR modules and their crystals. For example, it is sometimes possible to describe the classical highest weights of a KR module in terms of polytopes, and in type A there is a polytopal realization of KR crystals. I will explain some of the relationships between KR modules and Ehrhart theory which is the study of lattice points in polyhedra.