Most meshes are usually produced with both topological and geometrical irregularity (arbitrary valence, non-uniform sampling). This has been seen as a flaw hindering subsequent mesh processing, because most of the other signals we manipulate everyday (sound, image, video) are acquired and processed as regularly sampled data. Three-dimensional (3D) signals, be they surfaces or volumes, are however drastically and inherently different. Although the main body of work on mesh processing has focused on semi-regular meshes (on which the usual DSP tools can be extended quite nicely), we have focused on fully irregular meshes. Understanding this problem of irregularity, inherent to 3D sampling, is fundamental in widely different applications ranging from mesh modeling to smoothing, parameterization, remeshing, and to even compression or animation. In this talk, we will show some of our latest results (both theoretical and practical) and will also point to the remaining challenges.