The purpose of this paper is to give estimates on the number of conjugacy classes of maximal subgroups of the finite symmetric groups \(S_n\), on \(n\) letters in terms of \(n\). It is shown that this number is of the form \((\frac12+o(1))n\). The main work has to be done in establishing that \(S_n\) has at most \(n^{6/11+o(1)}\) conjugacy classes of primitive maximal subgroups. Of course, the O'Nan-Scott Theorem and the classification of finite simple groups are used. In the course of the proof, it is shown that any finite almost simple group has at most \(n^{17/11+o(1)}\) maximal subgroups of index \(n\). For most types of simple groups a slightly stronger estimate has been proved by \textit{A. Mann} and the second author [Isr. J. Math. 96, pt. B, 449-468 (1996)]; the remaining groups of Lie type \(F_4\), \(E^\varepsilon_6\), \(E_7\) and \(E_8\) are dealt with in the paper.