The main result of this paper is that every automorphism of a CAT(0) cube complex is semi-simple, meaning either combinatorially elliptic or combinatorially hyperbolic (up to possibly subdividing). The former means there is a fixed vertex, and the latter means there is a stabilized combinatorial geodesic on which the (positive) translation length is achieved. This is a rather strong result, and has some powerful consequences, for instance as soon as a finitely generated group contains a distorted cyclic subgroup, it cannot possibly act properly on a discrete space with walls. There are some groups of this sort, like Baumslag-Solitar groups and the Heisenberg group, that do nonetheless admit a proper action on a measured space with walls, so this establishes that acting properly on a measured space with walls is not sufficient to act properly on a discrete space with walls. It should be emphasized that there is no assumption of finite dimensionality on the CAT(0) cube complexes here.