For an almost simple, simply connected, connected algebraic subgroup \(G\) of \(\text{Gl}_m\) defined over the global field \(k\) let \(\Gamma=G(O_S)\) denote the corresponding \(S\)-arithmetic subgroup, \(S\) a finite set of valuations of \(k\) containing all the archimedean ones in the case of an algebraic number field \(k\), \(O_S\) the ring of \(S\)-integers of \(k\). Assume \(\Gamma\) to be infinite, or equivalently that \(\prod_{\nu\in S}G(k_\nu)\) is not compact. Let \(\sigma_n(\Gamma)\) (resp. \(\gamma_n(\Gamma)\)) be the number of all (resp. congruence) subgroups of index at most \(n\) in \(\Gamma\). The rate of growth of \(\sigma_n(\Gamma)\) (resp. \(\gamma_n(\Gamma)\)) and the links to the arithmetic of \(\Gamma\) are the subject of the paper under review. \(\Gamma\) is said to have the congruence subgroup property, if the congruence kernel \(\text{ker}(\widehat{G}(O_S)\to G(\widehat{O}_S))\) is finite (\(\widehat{\;}\) denotes the pro-finite completion), in which case \(\sigma_n(\Gamma)\) and \(\gamma_n(\Gamma)\) have the same type of growth. The main results are: (a) If \(\text{char}(k)=0\) then: \[ C_1{\log^2 n\over\log\log n}\leq\log\gamma_n(\Gamma)\leq C_2{\log^2 n\over\log\log n}\tag{i} \] for suitable constants \(C_1\) and \(C_2\). (ii) Assume \(G(k)\) has the standard description of normal subgroups, i.e. for every non-central finite index normal subgroup \(N\) of \(G(k)\) there exists an open normal finite index subgroup \(W \subseteq\prod_{\nu\in T}G(k_\nu)\) such that \(W\cap G(k)=N\), where \(T:=\{\nu\) finite place of \(k\mid G(k_\nu)\) is compact\} and \(T\) is disjoint from \(S\), and assume furthermore that \(G(k_\nu)\) is isotropic for every finite place \(v \in S\). Then \(\Gamma\) has the congruence subgroup property if and only if \(\text{log } \sigma_n (\Gamma)=o(\log^2 n)\). (iii) If under the assumptions of (ii) \(\Gamma\) is boundedly generated then \(\Gamma\) has the congruence subgroup property. (Here the group \(\Gamma\) has bounded generation, if there are elements \(g_1,\dots, g_l\) of \(\Gamma\) such that \(\Gamma=\langle g_1 \rangle \cdots \langle g_l\rangle\), \(\langle g_i\rangle\) the cyclic group generated by \(g_i\).) In the case of \(\text{char}(k) > 0\) the results are not as definitive as in the number field case, still the author shows that the rate of growth of \(\gamma_n (\Gamma)\) is different from the characteristic zero case, more precisely: (iv) Let \(\Gamma=G(O_S)\), assume that \(G\) splits over \(k\) and if \(\text{char} (k)=2\) then \(G\) is not of type \(A_1\) or \(C_n\). Then for suitable constants \(C_3\) and \(C_4\), \[ C_3 \log^2 n \leq \log \gamma_n (\Gamma) \leq C_4 \log^3 n \text{ holds.} \] The proof of the above results is by no means obvious: Firstly it uses a generalization of a classical result on \(\text{SL}_2 (Z)\) -- ``level \(\leq\) index'' -- to arbitrary \(\Gamma\). The problem of counting subgroups of an infinite group is then changed to counting subgroups of finite groups of type \(G(Z/mZ)\). This can be done by using a uniform version of the prime number theorem in arithmetic progressions, and parts of the classification theory of finite simple groups (Aschbacher and Guralnick).