In this paper we present a generalization of the Fast Marching method introduced by J. A. Sethian in 1996 to solve numerically the eikonal equation. The new method, named Buffered Fast Marching (BFM), is based on a semi-Lagrangian discretization and is suitable for Hamilton-Jacobi equations modeling monotonically advancing fronts, including Hamilton-Jacobi-Bellman and Hamilton-Jacobi- Isaacs equations which arise in the framework of optimal control problems and differential games. We also show the convergence of the algorithm to the viscosity solution. Finally we present several numerical tests proving that the BFM method is accurate and faster than the classical iterative algorithm in which every node of the grid is computed at every iteration.