This chapter establishes the main properties of the category of n-polygraphs. Limits and colimits are computed, and the category is proven to be complete and cocomplete. The behavior of the cartesian product deserves a special attention in that it does not correspond to the product of generators. The monomorphisms (resp. epimorphisms) in are then characterized as injective (resp. surjective) maps between generators. The linearization of polygraphic expressions plays a central role in proving these facts. Whereas the category of n-polygraph is a presheaf category in low dimensions, it already fails to be cartesian closed for n=3, the culprit for this defect being as usual the Eckmann-Hilton phenomenon. The categories of n-polygraph are, however, locally presentable. The technical notion of context is introduced in relation with n-dimensional rewriting, and used to prove that if an ω-category is freely generated by a polygraph, then this polygraph is unique up to isomorphism. Finally, rewriting properties of n-polygraphs are defined and coherence results are proven by rewriting on (n-1)-categories presented by convergent n-polygraphs.