[en] The theory of electron capture in multielectron ion-atom collisions based upon the atomic expansion method for collisions at a few keV/amu to a few MeV/amu is considered. Single-electron transfer processes are primarily considered. First-order perturbation theories for charge transfer, the K--K electron capture cross sections in heavy ion-atom collisions, the differential cross sections for electron transfer, electron capture from L- and M-subshells, quasi-molecular binding correction to the atomic expansion method, and further improvement of the theoretical model are treated. 25 references