Optimal transport (OT) has in recent years attracted great interest for its applications both within and outside of mathematics. The numerical solution of OT problems poses unique challenges and can be viewed from many different perspectives. We will review the formulation of OT problems, as well as the growing collection of approaches to solving them numerically, and then introduce a new method, which takes the perspective of nonlinear PDE. Our method rephrases Monge's OT problem with quadratic cost–via a Monge-Ampère equation–as an infinite-dimensional optimization problem, which is in fact a convex problem when the target is a log-concave measure with convex support. We define a natural finite-dimensional discretization to the problem, which can be solved via convex optimization, and discuss a proof of the convergence of the solutions of these optimization problems to the true solution of the OT problem. We then explain how to make this method more efficient and more accurate and how to remove the restriction on the target measure. (Joint work with Yanir Rubinstein)