The Dirac-Van Vleck-Serber spin-operator expansion, first applied by Anderson to the Kramers superexchange problem, is extended, simplified, and systematized in order to handle all overlap contributions arising from a number of interacting configurations. The linear cation-anion-cation (e.g., Mn++-O---Mn++) four-electron problem is worked out in detail, taking account of all contributions from configurations (A.) ionic, (B) electron transferred to right, (C) electron transferred to left. Group symmetry requirements are invoked; and these, together with a simple approximation equivalent to perturbation theory, are shown to reduce the complicated matrix formulation to a single linear equation. The solution contains terms previously obtained by Anderson, by Anderson and Hasegawa, and by Yamashita, and a number of important extra terms. All superexchange terms are fourth order or higher in the overlap S. A rough numerical evaluation with modified Slater wave functions appropriate to MnO-type crystals yields an effective superexchange integral of the required size. Brief consideration is given to configurations in which two electrons are transferred, in particular (D) simultaneous transfer of electrons to right and to left (Slater mechanism). Unless the energy required to form this configuration is surprisingly small, its contribution is probably not so important, although the problem needs to be investigated in detail. Some consideration is also given to the linear cation-anion-anion-cation (e.g., Mn++-Br--Br--Mn++) problem; the formal solution for the ionic configuration is worked out; and it is shown that superexchange terms first appear in the order S4T2, where S is the anion-cation overlap and T is the anion-anion overlap.