This thesis deals with the numerical modeling and simulation of granular media with large populations of non-spherical particles. Granular media are highly pervasive in nature and play an important role in technology. They are present in fields as diverse as civil engineering, food processing, and the pharmaceutical industry. For the physicist, they raise many challenging questions. They can behave like solids, as well as liquids or even gases and at times as none of these. Indeed, phenomena like granular segregation, arching effects or pattern formation are specific to granular media, hence often they are considered as a fourth state of matter. Around the turn of the century, the increasing availability of large computers made it possible to start investigating granular matter by using numerical modeling and simulation. Most numerical models were originally designed to handle spherical particles. However, making it possible to process non-spherical particles has turned out to be of utmost importance. Indeed, it is such grains that one finds in nature and many important phenomena cannot be reproduced just using spherical grains. This is the motivation for the research of the present thesis. Subjects in several fields are involved. The geometrical modeling of the particles and the simulation methods require discrete geometry results. A wide range of particle shapes is proposed. Those shapes, spheropolyhedra, are Minkowski sums of polyhedra and spheres and can be seen as smoothed polyhedra. Next, a contact detection algorithm is proposed that uses triangulations. This algorithm is a generalization of a method already available for spheres. It turns out that this algorithm relies on a positive answer to an open problem of computational geometry, the connectivity of the flip-graph of all triangulations. In this thesis it has been shown that the flip-graph of regular triangulations that share a same vertex set is connected. The modeling of contacts requires physics. Again the contact model we propose is based on the existing molecular dynamics model for contacts between spheres. Those models turn out to be easily generalizable to smoothed polyhedra, which further motivates this choice of particle shape. The implementation of those methods requires computer science. An implementation of this simulation methods for granular media composed of non-spherical particles was carried out based on the existing C++ code by J.-A. Ferrez that originally handled spherical particles. The resulting simulation code was used to gain insight into the behavior of granular matter. Three experiments are presented that have been numerically carried out with our models. The first of these experiments deals with the flowability (i. e. the ability to flow) of powders. The flowability of bidisperse bead assemblies was found to depend only on their mass-average diameters. Next, an experiment of vibrating rods inside a cylindrical container shows that under appropriate conditions they will order vertically. Finally, experiments investigating the shape segregation of sheres and spherotetrahedra are perfomed. Unexpectedly they are found to mix.