Title: Optimal Unsupervised Domain Translation. Abstract: Unsupervised Domain Translation~(UDT) consists in finding meaningful correspondences between two domains, without access to explicit pairings between them. Following the seminal work of \textit{CycleGAN}, many variants and extensions of this model have been applied successfully to a wide range of applications. However, these methods remain poorly understood, and lack convincing theoretical guarantees. In this work, we define UDT in a rigorous, non-ambiguous manner, explore the implicit biases present in the approach and demonstrate the limits of theses approaches. Specifically, we show that mappings produced by these methods are  biased towards \textit{low energy} transformations, leading us to cast UDT into an Optimal Transport~(OT) framework by making this implicit bias explicit. This not only allows us to provide theoretical guarantees for existing methods, but also to solve UDT problems where previous methods fail. Finally, making the link between the dynamic formulation of OT and CycleGAN, we propose a simple approach to solve UDT, and illustrate its properties in two distinct settings.