Level set and Fast Marching Methods are computational techniques for tracking the evolution of fronts moving under complex speed laws. Level set methods, introduced by Osher and Sethian, transform front propagation problems into an initial value PDE; the more recent Fast Marching Methods, introduced by Sethian, transform problems into a boudnary value PDE. Both rely on the theory of viscosity solutions of partial differential equations and schemes borrowed from the numerical solution of hyperbolic conservation laws. We will describe these techniques and their use in a large collection of applications, including optimal path planning in robotics, seismic analysis, etching and deposition in semi-conductor manufacturing, shape recovery in computer vision, combustion and fluid mechanics, grid generation and geometric singularities.