The purpose of this chapter is to introduce the notion of a polygraphic resolution of an ω-category. This notion was introduced by Métayer to define a homology theory for ω-categories, that is now known as the polygraphic homology. It was then showed by himself and Lafont that this homology recovers the classical homology of monoids for ω-categories coming from monoids. It is now known by work of Lafont, Métayer, and Worytkiewicz that these polygraphic resolutions are resolutions in the sense of a model category structure on ω-categories, the so-called folk model structure. Every ω-category is shown to admit such a resolution, and the relationship between two resolutions of the same ω-category is examined.