Domain Translation is the problem of finding a meaningful correspondence between two domains. Since in a majority of settings paired supervision is not available, much work focuses on Unsupervised Domain Translation (UDT) where data samples from each domain are unpaired... Following the seminal work of CycleGAN for UDT, many variants and extensions of this model have been proposed. However, there is still little theoretical understanding behind their success. We observe that these methods yield solutions which are approximately minimal w.r.t. a given transportation cost, leading us to reformulate the problem in the Optimal Transport (OT) framework. This viewpoint gives us a new perspective on Unsupervised Domain Translation and allows us to prove the existence and uniqueness of the retrieved mapping, given a large family of transport costs. We then propose a novel framework to efficiently compute optimal mappings in a dynamical setting. We show that it generalizes previous methods and enables a more explicit control over the computed optimal mapping. It also provides smooth interpolations between the two domains. Experiments on toy and real world datasets illustrate the behavior of our method. (read more)