The paper presents a fast sweeping method for solving the Eikonal equation on a rectangular grid. The method uses upwind difference for discretization and Gauss-Seidel iterations with alternating sweeping orderings to solve the discretized system. The key idea is that each sweeping ordering follows a family of characteristics in a certain direction. The method has an optimal complexity of O(N) for N grid points and is simple to implement in any dimension. Monotonicity and stability properties are proven, and convergence and error estimates for the distance function are studied. It is shown that 2^n Gauss-Seidel iterations are sufficient for the distance function in n dimensions. Numerical examples are used to verify the analysis. The method is optimal in the sense that a finite number of iterations is needed, and the complexity is O(N) for N grid points. The algorithm is extended to more general Hamilton-Jacobi equations. The fast sweeping method is compared with other methods, such as Danielsson's algorithm and a discrete approach proposed in [17]. The method is shown to be efficient and accurate for computing the distance function and general Eikonal equations. The paper also discusses the convergence and error estimates for the distance function and general Eikonal equations. The fast sweeping method is proven to converge monotonically to the solution of the discretized system. The error estimate for the distance function is shown to be O(h log(1/h)), and for general Eikonal equations, the numerical solution has the same accuracy as any other method using the same discretization. The number of iterations needed for the fast sweeping method depends on the right-hand side f(x) and the size and dimension of the computational domain. The method is shown to be efficient and accurate for computing the distance function and general Eikonal equations.The paper presents a fast sweeping method for solving the Eikonal equation on a rectangular grid. The method uses upwind difference for discretization and Gauss-Seidel iterations with alternating sweeping orderings to solve the discretized system. The key idea is that each sweeping ordering follows a family of characteristics in a certain direction. The method has an optimal complexity of O(N) for N grid points and is simple to implement in any dimension. Monotonicity and stability properties are proven, and convergence and error estimates for the distance function are studied. It is shown that 2^n Gauss-Seidel iterations are sufficient for the distance function in n dimensions. Numerical examples are used to verify the analysis. The method is optimal in the sense that a finite number of iterations is needed, and the complexity is O(N) for N grid points. The algorithm is extended to more general Hamilton-Jacobi equations. The fast sweeping method is compared with other methods, such as Danielsson's algorithm and a discrete approach proposed in [17]. The method is shown to be efficient and accurate for computing the distance function and general Eikonal equations. The paper also discusses the convergence and error estimates for the distance function and general Eikonal equations. The fast sweeping method is proven to converge monotonically to the solution of the discretized system. The error estimate for the distance function is shown to be O(h log(1/h)), and for general Eikonal equations, the numerical solution has the same accuracy as any other method using the same discretization. The number of iterations needed for the fast sweeping method depends on the right-hand side f(x) and the size and dimension of the computational domain. The method is shown to be efficient and accurate for computing the distance function and general Eikonal equations.