We study a variant of the capacitated vehicle routing problem (CVRP), which asks for thecost-optimal delivery of a single product to geographically dispersed customers through a fleetof capacity-constrained vehicles. Contrary to the classical CVRP, which assumes that the cus-tomer demands are deterministic, we model the demands as a random vector whose distributionis only known to belong to an ambiguity set. We then require the delivery schedule to be feasi-ble with a probability of at least 1− , where characterizes the risk tolerance of the decisionmaker. We show that the emerging distributionally robust CVRP can be solved with standardbranch-and-cut algorithms whenever the ambiguity set satisfies a subadditivity condition. Wethen argue that this subadditivity condition holds for a large class of moment ambiguity sets.We derive cut generation schemes for ambiguity sets that specify the support as well as (boundson) the first and second moments of the customer demands. Our numerical results indicate thatthe distributionally robust CVRP has favorable scaling properties and can often be solved inruntimes comparable to those of the deterministic CVRP.