This paper introduces a new notion of solution for first-order Hamilton-Jacobi equations (HJ equations), called viscosity solutions. These solutions allow for functions that are nowhere differentiable but still satisfy strong uniqueness, stability, and existence theorems. The paper focuses on two types of problems: the Dirichlet problem, where the equation is solved with given boundary conditions, and the Cauchy problem, where the equation is solved with given initial conditions. The key idea is to define a solution in terms of test functions and their interactions with the Hamiltonian, ensuring that the solution satisfies the equation in a weak sense. The paper establishes that viscosity solutions are consistent with classical solutions when they exist and provides existence and uniqueness results for both the Dirichlet and Cauchy problems. It also shows that viscosity solutions are stable under convergence and can be used in numerical approximation methods. The paper concludes with a discussion of the broader implications of viscosity solutions for the study of Hamilton-Jacobi equations.This paper introduces a new notion of solution for first-order Hamilton-Jacobi equations (HJ equations), called viscosity solutions. These solutions allow for functions that are nowhere differentiable but still satisfy strong uniqueness, stability, and existence theorems. The paper focuses on two types of problems: the Dirichlet problem, where the equation is solved with given boundary conditions, and the Cauchy problem, where the equation is solved with given initial conditions. The key idea is to define a solution in terms of test functions and their interactions with the Hamiltonian, ensuring that the solution satisfies the equation in a weak sense. The paper establishes that viscosity solutions are consistent with classical solutions when they exist and provides existence and uniqueness results for both the Dirichlet and Cauchy problems. It also shows that viscosity solutions are stable under convergence and can be used in numerical approximation methods. The paper concludes with a discussion of the broader implications of viscosity solutions for the study of Hamilton-Jacobi equations.