Given two d-dimensional probability distribution A and B, the optimal transport (OT) problem asks for computing a minimum-cost plan to transporting the mass of A to B. The cost of the OT plan is referred to as the Wasserstein or earth-mover’s distance. This talk presents efficient approximation algorithms for computing an OT plan, and its variants, for both discrete and continuous distributions. It also discusses algorithms for computing the Wasserstein barycenter.