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The Hamilton-Jacobi equation arises in a variety of applications such as optimal control, game theory, geometric optics, front propagation and finance. The Hamilton-Jacobi equation is also one of the formulation of classical mechanics in which the motion of a particle can be represented as a wave. This book involves the theoretical aspects of the Hamilton-Jacobi equation and its associated system of conservation law with their numerical simulation. First, the existence and uniqueness of the solution of conservation law is considered in weak sense. This idea is used to relate the existence and uniqueness for the Hamilton-Jacobi equations as it is a particular case of conservation law. The classical numerical schemes and their properties are introduced for the purpose of numerical experiments. An application of this in finance through Black-Scholes equation is considered.