This article, presented by Jean-Michel Lasry and Pierre-Louis Lions, surveys recent studies on mean-field models, inspired by Statistical Mechanics and Physics. The authors focus on three examples from Economics and Finance, where a large number of "rational players" with limited information make optimal strategies based on global or macroscopic information. These scenarios lead to nonlinear differential equations, which are new types of problems. The main goal is to study these equations, establish their well-posedness, and explore their connections with various fields of Analysis. The article also discusses limiting cases, examples, possible extensions, and open problems. The work is based on the 1st Takagi Lectures delivered at Kyoto University in 2006.This article, presented by Jean-Michel Lasry and Pierre-Louis Lions, surveys recent studies on mean-field models, inspired by Statistical Mechanics and Physics. The authors focus on three examples from Economics and Finance, where a large number of "rational players" with limited information make optimal strategies based on global or macroscopic information. These scenarios lead to nonlinear differential equations, which are new types of problems. The main goal is to study these equations, establish their well-posedness, and explore their connections with various fields of Analysis. The article also discusses limiting cases, examples, possible extensions, and open problems. The work is based on the 1st Takagi Lectures delivered at Kyoto University in 2006.