The article provides a comprehensive introduction to the theory of viscosity solutions for second-order fully nonlinear partial differential equations. The authors, Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, present the basic theory in a self-contained manner, covering key concepts, existence and uniqueness theorems, and applications. The theory allows for continuous functions to be solutions of such equations, providing a flexible framework that generalizes classical solutions. The introduction highlights the importance of degenerate ellipticity and properness conditions, which are essential for the theory's applicability. The article includes numerous examples to illustrate the theory's broad scope and practical applications, such as Hamilton-Jacobi-Bellman equations and Monge-Ampère equations. The authors also discuss the maximum principle for semicontinuous functions and comparison results for the Dirichlet problem, laying the groundwork for further developments in the theory.The article provides a comprehensive introduction to the theory of viscosity solutions for second-order fully nonlinear partial differential equations. The authors, Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, present the basic theory in a self-contained manner, covering key concepts, existence and uniqueness theorems, and applications. The theory allows for continuous functions to be solutions of such equations, providing a flexible framework that generalizes classical solutions. The introduction highlights the importance of degenerate ellipticity and properness conditions, which are essential for the theory's applicability. The article includes numerous examples to illustrate the theory's broad scope and practical applications, such as Hamilton-Jacobi-Bellman equations and Monge-Ampère equations. The authors also discuss the maximum principle for semicontinuous functions and comparison results for the Dirichlet problem, laying the groundwork for further developments in the theory.