Over the past few years, optimal transport has gained popularity in machine learning as a way to compare probability distributions. Unlike more classical dissimilarities for probability measures, such as the Kullback-Leibler divergence, optimal transport distances (or Wasserstein distances) can deal with distributions of disjoint supports by taking into account the geometry of the underlying ground space. This strength is, however, hampered by the fact that these distances are usually computed by solving a linear program, resulting, when this ground space is high-dimensional, in well documented statistical challenges, usually referred to as the ``curse'' of dimensionality. Finding new methodologies that can mitigate this issue is therefore crucial if one wants optimal transport-based algorithms to perform well on real data.Beyond this purely metric aspect, another appealing feature of optimal transport theory is that it provides mathematical tools to study maps that are able to morph (or push-forward) a measure into another. Such maps are playing an increasingly important role in various areas of science (biology, neuroimaging) or subdomains in machine learning (generative models, domain adaptation), to name a few. Estimating such morphings, or maps, that are both optimal and able to generalize outside the data, is an open problem.In this thesis, we propose a new estimation framework to compute proxies to the Wasserstein distance. That framework aims at handling high-dimensionality by taking advantage of the low-dimensional structures hidden in the distributions. This can be achieved by projecting the measures onto a subspace chosen so as to maximize the Wasserstein distance between their projections. In addition to this novel methodology, we show that this framework falls into a broader connection between regularization when computing Wasserstein distances and adversarial robustness.In the next contribution, we start from the same problem, estimation of optimal transport in high dimensions, but adopt a different perspective: rather than changing the ground cost, we go back to the more fundamental Monge perspective on optimal transport and use the Brenier theorem and Caffarelli's regularity theory to propose a new estimation procedure to characterize maps that are Lipschitz and gradients of strongly convex functions.