The progress in solution of problems involving non-relativistic fast ion-atom collisions with two actively participating electrons has been reviewed. Leading quantum mechanical methods have been analyzed in the framework of four-body scattering theory. A simple formalism has been described on the basis of the Dodd-Greider integral equations for a four-particle system, which were used to analyze the two-electron charge-exchange process at medium and high velocities of relative motion of particles. An important feature of the developed four-particle continuum distorted waves (CDW) method is the consistent preservation of the proper asymptotic limits of the wave functions of a colliding system in the entrance and exit channels of the reaction, which takes into account the long-range nature of Coulomb interactions. The amplitude of the two-electron charge-exchange reaction has been calculated in the approximation of the mechanism of simultaneous capture by an incident particle of two target electrons. The calculations using the presented theory have been made on the example of a two-electron capture reaction in high-energy He 2+ +He collisions.