The mathematical theory of Optimal Transport (OT) is now one of the most visible and also most active fields in nonlinear analysis, with two Fieldsmedal winners Cedric Villani and Alessio Figalli among its protagonists. The origins of OT date back to the works of GaspardMonge in the 18th century and those of Leonid Kantorovich in the ’1940s. However, what is considered as the foundations of the modern theory have mostly been established in several ground-breaking works at the end of the 20th century: among the most prominent contributors, there is Yann Brenier, who characterized optimal transport in terms of connections with PDEs and hydrodynamics in the 1980’s; there is RobertMcCann, who introduced the powerful concept of displacement convexity in the 1990’s; there is Felix Otto, who demonstrated the implications of OT to nonlinear evolution equations around the millennium.