Optimal transport (OT) provides effective tools for comparing and mapping\nprobability measures. We propose to leverage the flexibility of neural networks\nto learn an approximate optimal transport map. More precisely, we present a new\nand original method to address the problem of transporting a finite set of\nsamples associated with a first underlying unknown distribution towards another\nfinite set of samples drawn from another unknown distribution. We show that a\nparticular instance of invertible neural networks, namely the normalizing\nflows, can be used to approximate the solution of this OT problem between a\npair of empirical distributions. To this aim, we propose to relax the Monge\nformulation of OT by replacing the equality constraint on the push-forward\nmeasure by the minimization of the corresponding Wasserstein distance. The\npush-forward operator to be retrieved is then restricted to be a normalizing\nflow which is trained by optimizing the resulting cost function. This approach\nallows the transport map to be discretized as a composition of functions. Each\nof these functions is associated to one sub-flow of the network, whose output\nprovides intermediate steps of the transport between the original and target\nmeasures. This discretization yields also a set of intermediate barycenters\nbetween the two measures of interest. Experiments conducted on toy examples as\nwell as a challenging task of unsupervised translation demonstrate the interest\nof the proposed method. Finally, some experiments show that the proposed\napproach leads to a good approximation of the true OT.\n