The progress in solving problems involving nonrelativistic fast ion (atom)-atom collisions with two actively participating electrons is reviewed. Such processes involve, e.g., (i) scattering between a bare nucleus (projectile) $P$ of charge $Z_P$ and a heliumlike atomic system consisting of two electrons $e_1$ and $e_2$ initially bound to the target nucleus $T$ of charge $Z_T$, i.e., the $Z_P\text{−}{(Z_T;e_1,e_2)}_i$ collisions; (ii) scattering between two hydrogenlike atoms ${(Z_P,e_1)}_{i_1}$ and ${(Z_T,e_2)}_{i_2}$, etc. A proper description of these collisional processes requires solutions of four-body problems with four active particles including two nuclei and two electrons. Among various one- as well as two-electron transitions which can occur in such collisions, special attention will be paid to double-electron capture, simultaneous transfer and ionization, simultaneous transfer and excitation, single-electron detachment and single-electron capture. Working within the four-body framework of scattering theory and imposing the proper Coulomb boundary conditions on the entrance and exit channels, the leading quantum-mechanical theories are analyzed. Both static and dynamic interelectron correlations are thoroughly examined. The correct links between scattering states and perturbation potentials are strongly emphasized. Selection of the present illustrations is dictated by the importance of interdisciplinary applications of energetic ion-atom collisions, ranging from thermonuclear fusion to medical accelerators for hadron radiotherapy.

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