Appears in collection : Francophone Computer Algebra Days 2025 / Journées nationales de calcul formel 2025

This course is dedicated to core algorithms of polyhedral geometry and covers theoretical aspects as well as practical ones. We will start with rational polyhedra, their projections and the conversions between their different types of representations. We will continue with a tour of the different questions related to the lattice points of rational polyhedra : checking existence, counting these points, describing them, in particular for the case of parametric polyhedra. Practical applications of rational polyhedra and lattice polyhedra require, at least in theory, to perform quantifier elimination, we will see how this is done in the context of optimizing compilers.