This paper presents a fast sweeping method for solving Eikonal equations on a rectangular grid. The method is an iterative approach that uses upwind differences 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 specific direction simultaneously. The method has an optimal complexity of \(O(N)\) for \(N\) grid points and is simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven, and convergence and error estimates for the distance function are studied in detail. It is shown that \(2^n\) Gauss-Seidel iterations are sufficient for the distance function in \(n\) dimensions. An estimation of the number of iterations for general Eikonal equations is also discussed. Numerical examples are provided to verify the analysis.This paper presents a fast sweeping method for solving Eikonal equations on a rectangular grid. The method is an iterative approach that uses upwind differences 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 specific direction simultaneously. The method has an optimal complexity of \(O(N)\) for \(N\) grid points and is simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven, and convergence and error estimates for the distance function are studied in detail. It is shown that \(2^n\) Gauss-Seidel iterations are sufficient for the distance function in \(n\) dimensions. An estimation of the number of iterations for general Eikonal equations is also discussed. Numerical examples are provided to verify the analysis.