  Optimal Transport has received much attention in Machine Learning as it
allows to compare probability distributions by exploiting the geometry of the
underlying space. However, in its original formulation, solving this problem
suffers from a significant computational burden. Thus, a meaningful line of
work consists at proposing alternatives to reduce this burden while still
enjoying its properties. In this thesis, we focus on alternatives which use
projections on subspaces. The main such alternative is the Sliced-Wasserstein
distance, which we first propose to extend to Riemannian manifolds in order to
use it in Machine Learning applications for which using such spaces has been
shown to be beneficial in the recent years. We also study sliced distances
between positive measures in the so-called unbalanced OT problem. Back to the
original Euclidean Sliced-Wasserstein distance between probability measures, we
study the dynamic of gradient flows when endowing the space with this distance
in place of the usual Wasserstein distance. Then, we investigate the use of the
Busemann function, a generalization of the inner product in metric spaces, in
the space of probability measures. Finally, we extend the subspace detour
approach to incomparable spaces using the Gromov-Wasserstein distance.
