This chapter is dedicated to the definition of 2-polygraphs, which are a 2-dimensional generalization of 1-polygraphs. Before introducing this notion, a refined viewpoint over 1-polygraphs is given. Instead of merely focusing on the set presented by a 1-polygraph as a set of equivalence classes of generators modulo the relations, the free category generated by the polygraph is now considered. The notion of 2-polygraph naturally appears as soon as arbitrary, non necessarily free, small categories are considered. In order to present such a category, one starts with a polygraph such that the 1-generators generate the morphisms of the category, but now it must be taken into account the relations induced by the category among the morphisms of the free category generated the resulting 1-polygraph. These relations will be generated by a set of 2-generators, consisting in certain pairs of morphisms intended to be equalized in the category. Following the same pattern, it will be explained that a 2-polygraph can also be seen as a system of generators for a free 2-category, thus preparing the study of 3-polygraphs. The variant where a (2,1)-category is freely generated is also examined.