This chapter introduces the notion of acyclic extension of a 2-category, which consists of the additional data of 3-generators "filling all the spheres". This leads to the notion of coherent presentation of a category, which consists of a 2-polygraph presenting the category together with an acyclic extension of the free (2,1)-category on the polygraph. Coherent presentations are then constructed from convergent ones, and the appropriate notion of Tietze transformation between coherent presentations is studied: this allows formulation of a coherent variant of the Knuth-Bendix completion procedure, but also a reduction procedure, which can be used to obtain smaller coherent presentations. Finally, coherent presentations of algebras are studied, thereby defining the proper notion of coherent extension for linear polygraphs.