Generative Adversarial Networks (GANs) have proven to be a powerful framework for learning to draw samples from complex distributions. In this talk I will discuss the challenge of learning sequential data via GANs. This notably requires the choice of a loss function that reflects the discrepancy between (measures on) paths. To take on this task, we employ adapted versions of optimal transport distances, that result from imposing a temporal causality constraint on classical transport problems. This constraint provides a natural framework to parameterize the cost function that is learned by the discriminator as a robust (worst-case) distance. We then employ a modification of the empirical measure, to ensure consistency of the estimators. Following Genevay et al. (2018), we also include an entropic penalization term which allows for the use of the Sinkhorn algorithm when computing the optimal transport cost.