What can one say about the function \(f(p,n)\) that counts (up to isomorphism) groups of order \(p^n\), where \(p\) is a prime, and \(n\) is an integer? \textit{G. Higman} [Proc. Lond. Math. Soc. (3) 10, 24-30 (1960; Zbl 0093.02603)] and \textit{C. C. Sims} [Proc. Lond. Math. Soc. (3) 15, 151-166 (1965; Zbl 0133.28401)] have given an asymptotic formula for fixed \(p\), and growing \(n\). Moreover, \textit{G. Higman} has formulated [Proc. Lond. Math. Soc. (3) 10, 566-582 (1960; Zbl 0201.36502)] the following PORC conjecture: given \(n\), there are an integer \(N\) and polynomials \(P_{n,i}(X)\), for \(0\leq i\leq N-1\), such that if \(p\equiv i\pmod N\), then \(f(n,p)=P_{n,i}(p)\). (PORC stands Polynomial On Congruence Classes.) In the important paper under review, the author introduces the function \(f(n,p,c,d)\) that counts groups of order \(p^n\), nilpotency class \(c\), generated by at most \(d\) generators. He proves that this function satisfies a linear recurrence relation with constant coefficients. This he does by proving that a zeta function naturally built up from the latter function \(f\) is rational. The ``smooth behaviour`` of this function \(f\), in the appropriate words of the author, shows a surprising regularity in the distribution of finite \(p\)-groups. The author then shows that counting finite \(p\)-groups amounts to counting points on varieties mod \(p\); moreover, these varieties are explicitly defined and arise from the resolution of singularities of a polynomial associated to each pair \((c,d)\) as above. In the second part of the paper, the author considers certain trees, whose vertices are \(p\)-groups, that have become natural objects of study with the advent of the coclass theory for (pro-)\(p\)-groups. (See for instance the book by \textit{C. R. Leedham-Green} and \textit{S. McKay} [The structure of groups of prime power order (London Mathematical Society Monographs. New Series. 27. Oxford: Oxford University Press) (2002; Zbl 1008.20001)].) He is able to prove that when the twigs of these trees are all pruned at length \(M\) (see the Introduction of the paper for the details), then these trees become ultimately periodic. This confirms the qualitative part of Conjecture P of \textit{M. F. Newman} and \textit{E. A. O'Brien} [Trans. Am. Math. Soc. 351, 131-169 (1999; Zbl 0914.20020)]. This result is obtained by proving the rationality of a certain function. The deep proof makes use of model theory, and of the theory of definable \(p\)-adic integrals. In the words of the author, ``the philosophy of this paper is that [\dots] zeta functions are a very powerful tool in understanding finite \(p\)-groups and nilpotent groups.'' He hopes that this paper will ``stimulate group theorists to adopt these zeta functions as a potentially very powerful weapon in their Arsenal.'' The striking and surprising results of this paper certainly prove this point beyond any doubt, although some readers may wish to substitute their own favourite football team to the one notoriously supported by the author.