Optimal transport originally refers to the following problem. Suppose that moving some pile of soil has a cost proportional to the mass of the pile and the undergone distance. How to transport a certain total mass of soil from a set of sources to a set of destinations, while minimizing the total cost? The question was first asked formally by Monge in the 1790s. Monge searched the optimal assignment of a destination to each source, and observed surprising geometric aspects of the problem. The problem was rediscovered by the economist Kantorovich, who pioneered linear optimization in the 1930s and generalized the Monge problem through the representation of the mass at the source and at the destination by probability distributions. The Kantorovich formulation proved powerful and spawned a vast number of research questions. The work of Brenier, Gangbo, and McCann in the 1990’s revived the topic, and strong connections were progressively found with central questions in mathematics, such as in dynamical systems, fluid mechanics, partial differential equations, and probability theory. Meanwhile, efforts to solve the optimization problem numerically led to a variety of computational advances and applications in econometrics, statistics, machine learning, vision, and genomics.