The paper discusses the metal-insulator transition in a weakly interacting many-electron system with localized single-particle states in disordered conductors. The authors prove that in the absence of coupling to an external bath, the dc electrical conductivity vanishes at low temperatures up to a critical temperature \( T_c \), while it becomes finite at higher temperatures. This implies a finite-temperature metal-insulator transition, where the many-body wave functions in the Fock space are Anderson-like localized. The metallic and insulating states are not distinguished by spatial or discrete symmetries. The authors formulate an effective Hamiltonian description at low energies and use quantum Boltzmann equations to determine kinetic coefficients in the metallic phase. In the insulating phase, they apply the Feynmann diagram technique to find the probability distribution function for quantum-mechanical transition rates. They show that the probability of an escape rate from a given quantum state to be finite vanishes in every order of perturbation theory, indicating that electron-electron interaction alone cannot cause relaxation or thermal equilibrium. When a weak coupling to a bath is introduced, the conductivity becomes finite even in the insulating phase, and it is much larger than phonon-induced hopping conductivity at high temperatures. This enhancement is attributed to the gradual stability loss of the insulating state as the transition temperature is approached, leading to a cascade of electronic hops. The paper also discusses the macroscopic manifestations of the many-body localization transition, including the existence of an extensive many-body mobility edge \(\mathcal{E}_c \propto \mathcal{V}\). The proof of the existence of this transition is based on the validity of high-temperature expansions for quantum corrections to conductivity and the correspondence between the many-electron interacting system and the Anderson model on a graph. The authors provide a detailed analysis of the microscopic mechanism of the many-body localization transition, including the estimation of the connectivity \(K\) and the structure of the matrix elements.The paper discusses the metal-insulator transition in a weakly interacting many-electron system with localized single-particle states in disordered conductors. The authors prove that in the absence of coupling to an external bath, the dc electrical conductivity vanishes at low temperatures up to a critical temperature \( T_c \), while it becomes finite at higher temperatures. This implies a finite-temperature metal-insulator transition, where the many-body wave functions in the Fock space are Anderson-like localized. The metallic and insulating states are not distinguished by spatial or discrete symmetries. The authors formulate an effective Hamiltonian description at low energies and use quantum Boltzmann equations to determine kinetic coefficients in the metallic phase. In the insulating phase, they apply the Feynmann diagram technique to find the probability distribution function for quantum-mechanical transition rates. They show that the probability of an escape rate from a given quantum state to be finite vanishes in every order of perturbation theory, indicating that electron-electron interaction alone cannot cause relaxation or thermal equilibrium. When a weak coupling to a bath is introduced, the conductivity becomes finite even in the insulating phase, and it is much larger than phonon-induced hopping conductivity at high temperatures. This enhancement is attributed to the gradual stability loss of the insulating state as the transition temperature is approached, leading to a cascade of electronic hops. The paper also discusses the macroscopic manifestations of the many-body localization transition, including the existence of an extensive many-body mobility edge \(\mathcal{E}_c \propto \mathcal{V}\). The proof of the existence of this transition is based on the validity of high-temperature expansions for quantum corrections to conductivity and the correspondence between the many-electron interacting system and the Anderson model on a graph. The authors provide a detailed analysis of the microscopic mechanism of the many-body localization transition, including the estimation of the connectivity \(K\) and the structure of the matrix elements.