The main purpose of this chapter is to give a proof of the celebrated theorem of Golod and Šafarevič, which gives an accurate lower bound for the minimal number r'(G) of relations needed to define a finite p-group G minimally generated by d(G) elements. The naive bound r'(G) ≥ d(G) of Theorem 6.7 is relegated to the humble role of a lemma, to be invoked almost unconsciously in the penultimate line of the proof. The proof we give is due to P. Roquette and is extremely elegant, modulo the rather technical machinery needed to begin it. We shall need the notions of a Gmodule A, and of the cohomology groups Hn (G,A), n ∈ No. If the field k of p-elements is made into a G-module in a trivial way, it turns out that H1 (G,k) is a vector space over k of dimension d(G), while the dimension r(G) of H2 (G,k) is at most r'(G). The last fact is proved in Theorem 20.3 using an argument based on the presentation theory of group extensions. Thus we begin with an account of the classical theory of group extensions, and then proceed to establish the connection with group cohomology. The only remaining preliminaries are a localization process and some basic facts about finite p-groups. We conclude by shedding a little light on the unsolved problem of classifying those finite p-groups for which r(G) and r'(G) are equal.