This review treats the process of dissociative recombination of molecular ions with electrons and its role in shaping the properties of a weakly-ionized plasma of the inert gases and mixtures of them. Data are presented on the rate constants of the process and the efficiency of recombination with formation of excited atoms in various states. We discuss experimental methods that enable study of recombination processes over a wide range of variation of the electron temperature. We analyze the competition of dissociative recombination with alternative mechanisms of formation of excited atoms in recombinatively nonequilibrium plasmas, the manifestation of heteronuclear ions in the kinetics of excited and charged particles, and the mechanisms of relaxation of the energy of vibrational motion in collisions of molecular ions with atoms and dissociation in collisions with electrons. The results are discussed of experiments to study the role of inelastic atom-atom collisions in the formation of the emission spectrum of the plasma with molecular ions.