Jean-Marie Mirebeau - Université d'Orsay The fast marching method is a numerical technique for computing distance maps on manifolds, by solving eikonal partial differential equations. It was originally developped in the context of conformal metrics, which locally proportionnal to the euclidean norm. Numerous applications, such as seismology, motion planning, or image processing, however require to go beyond this special case and to consider Riemannian, Finslerian, or even degenerate metrics. I will present some adaptations of the method to these contexts, designed leveraging tools from discrete geometry, and their applications.