This paper introduces a new notion of solution for Hamilton-Jacobi equations, which are first-order nonlinear partial differential equations. The authors focus on two types of problems: the Dirichlet problem, where the equation is given on the boundary of a domain, and the Cauchy problem, where the equation is given in a domain with initial conditions. Classical solutions to these problems are often not well-behaved due to the crossing of characteristics, leading to limitations in their analysis. To address this, the authors propose a new concept called "viscosity solutions." Viscosity solutions allow for functions that are not differentiable everywhere but still satisfy the equation in a weak sense. This notion enables strong uniqueness theorems, stability theorems, and general existence theorems for these problems. The paper provides detailed definitions and basic properties of viscosity solutions, including stability results and characterizations of points in the domain of the solution. The authors also discuss the uniqueness of viscosity solutions for both the Dirichlet and Cauchy problems, showing that if \( u \) and \( v \) are viscosity solutions of the same problem, then \( u \equiv v \). They further explore the implications of viscosity solutions in numerical approximation and provide convergence theorems for difference approximations. The paper concludes with a discussion on changes of variables and differential inequalities in the viscosity sense, providing additional tools for analyzing viscosity solutions. Overall, the work establishes a robust framework for understanding and solving Hamilton-Jacobi equations, even in cases where classical solutions do not exist.This paper introduces a new notion of solution for Hamilton-Jacobi equations, which are first-order nonlinear partial differential equations. The authors focus on two types of problems: the Dirichlet problem, where the equation is given on the boundary of a domain, and the Cauchy problem, where the equation is given in a domain with initial conditions. Classical solutions to these problems are often not well-behaved due to the crossing of characteristics, leading to limitations in their analysis. To address this, the authors propose a new concept called "viscosity solutions." Viscosity solutions allow for functions that are not differentiable everywhere but still satisfy the equation in a weak sense. This notion enables strong uniqueness theorems, stability theorems, and general existence theorems for these problems. The paper provides detailed definitions and basic properties of viscosity solutions, including stability results and characterizations of points in the domain of the solution. The authors also discuss the uniqueness of viscosity solutions for both the Dirichlet and Cauchy problems, showing that if \( u \) and \( v \) are viscosity solutions of the same problem, then \( u \equiv v \). They further explore the implications of viscosity solutions in numerical approximation and provide convergence theorems for difference approximations. The paper concludes with a discussion on changes of variables and differential inequalities in the viscosity sense, providing additional tools for analyzing viscosity solutions. Overall, the work establishes a robust framework for understanding and solving Hamilton-Jacobi equations, even in cases where classical solutions do not exist.