This article presents a self-contained exposition of the theory of viscosity solutions for second-order fully nonlinear partial differential equations (PDEs). The theory provides a framework for proving comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence for a wide range of PDEs. The key idea is to define solutions in terms of upper and lower semicontinuous functions and their associated second-order generalized jets, which allow for the treatment of equations that may not have classical smooth solutions. The theory applies to equations of the form $ F(x, u, Du, D^2u) = 0 $, where $ F $ satisfies a monotonicity condition. This condition ensures that the equations are "degenerate elliptic" and allows for the definition of viscosity solutions. The article discusses various examples of such equations, including the Laplace equation, degenerate elliptic linear equations, first-order equations, quasilinear elliptic equations, Hamilton-Jacobi-Bellman equations, and Monge-Ampère equations. These examples illustrate the broad applicability of the theory. The article introduces the concept of viscosity solutions, which are defined using test functions and generalized jets. The theory is then used to prove comparison principles, existence results, and other important properties for PDEs. The maximum principle for semicontinuous functions is discussed, and the article shows how to use Perron's method to establish existence of solutions. The theory is also extended to handle generalized boundary conditions and problems with non-strict boundary conditions. The article concludes with a discussion of parabolic problems, singular equations, and applications of the theory. It also includes an appendix that provides a self-contained proof of a key result in the theory. The paper emphasizes the importance of the theory in various areas of mathematics, including optimal control, differential games, and calculus of variations. The theory is presented in a clear and accessible manner, with detailed examples and explanations.This article presents a self-contained exposition of the theory of viscosity solutions for second-order fully nonlinear partial differential equations (PDEs). The theory provides a framework for proving comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence for a wide range of PDEs. The key idea is to define solutions in terms of upper and lower semicontinuous functions and their associated second-order generalized jets, which allow for the treatment of equations that may not have classical smooth solutions. The theory applies to equations of the form $ F(x, u, Du, D^2u) = 0 $, where $ F $ satisfies a monotonicity condition. This condition ensures that the equations are "degenerate elliptic" and allows for the definition of viscosity solutions. The article discusses various examples of such equations, including the Laplace equation, degenerate elliptic linear equations, first-order equations, quasilinear elliptic equations, Hamilton-Jacobi-Bellman equations, and Monge-Ampère equations. These examples illustrate the broad applicability of the theory. The article introduces the concept of viscosity solutions, which are defined using test functions and generalized jets. The theory is then used to prove comparison principles, existence results, and other important properties for PDEs. The maximum principle for semicontinuous functions is discussed, and the article shows how to use Perron's method to establish existence of solutions. The theory is also extended to handle generalized boundary conditions and problems with non-strict boundary conditions. The article concludes with a discussion of parabolic problems, singular equations, and applications of the theory. It also includes an appendix that provides a self-contained proof of a key result in the theory. The paper emphasizes the importance of the theory in various areas of mathematics, including optimal control, differential games, and calculus of variations. The theory is presented in a clear and accessible manner, with detailed examples and explanations.