Explore free boundary problems within the context of optimal transportation theory in this 51-minute lecture delivered at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the mathematical complexities where optimal transport meets free boundary analysis, examining how these problems arise naturally in transportation theory and their theoretical implications. Learn about the intersection of partial differential equations, calculus of variations, and geometric analysis as applied to optimal transport scenarios. Gain insights into current research developments and mathematical techniques used to address free boundary challenges in this specialized area of mathematics, presented as part of the comprehensive Thematic Programme on Free Boundary Problems.