Anick and Green constructed the first explicit free resolutions for algebras from a presentation of relations by non-commutative Gröbner bases, which allow computing homological invariants, such as homology groups, Hilbert and Poincaré series of algebras presented by generators and relations given by a Gröbner basis. Similar methods for calculating free resolutions for monoids and algebras, inspired by string rewriting mechanisms, have been developed in numerous works. A purely polygraphic approach to the construction of these resolutions by rewriting has been developed using the notion of (ω,1)-polygraphic resolution, where the mechanism for proving the acyclicity of the resolution relies on the construction of a normalization strategy extended in all dimensions. The construction of polygraphic resolutions by rewriting has also been applied to the case of associative algebras and shuffle operads, introducing in each case a notion of polygraph adapted to the algebraic structure. This chapter demonstrates how to construct a polygraphic resolution of a category from a convergent presentation of that category, and how to deduce an abelian version of such a resolution.