The author describes the solvable subgroups of maximal order in the symmetric and alternating groups, and in the general and special linear groups over a finite field. The maximal order of a solvable subgroup in the symmetric groups was discussed by the reviewer [in J. Aust. Math. Soc. 7, 417-424 (1967; Zbl 0153.040)] and in the general linear groups by \textit{D. A. Suprunenko} [Matrix Groups (1976; see Zbl 0253.20074 for a review)] and \textit{T. R. Wolf} [Can. J. Math. 5, 1097-1111 (1982; Zbl 0476.20028)]. In the present paper, the author shows that, for the symmetric and alternating groups, the subgroups in question are all conjugate and may be constructed via an iterated wreath product. For the linear groups, the subgroups in question are all conjugate to the subgroup of all upper triangular matrices, except in some cases where the field has 2, 3, 5 or 7 elements.