How can the mass of a reference distribution be rearranged to match a target distribution, subject to a minimal transportation cost? This problem was first formulated in the realm of logistics and measure transport is nowadays an indispensable tool in modern applications such as image retrieval, inverse problems, cancer detection, shape and image registration, and machine learning. This class will introduce the foundations of measure transport at the intersection of analysis, probability and optimization, its applications and connections to various fields and explore modern computational methods.