A molecular theory of the dynamics of ions and solvent molecules in electrolyte solutions is presented, The theory properly includes ion-ion, ion-solvent, and solvent-solvent molecular correlations through intra- and interspecies static structure factors and direct correlation functions. Both diffusive and nondiffusive (such as inertial) modes of relaxation of ions and solvent molecules are included in the theory. Explicit results are obtained for the time dependence of ion-ion, ion-solvent, and solvent-solvent van Hove functions at zero and finite wave vectors for solutions of varying ion concentration and dipolar strength. Frequency- and wave vector-dependent dynamic response functions of electrolyte solutions are also calculated by employing linear response theory. It is found that the dynamic response of ions and solvent molecules at finite wave vectors can be very different from that at zero wave vector (or at long wavelength). An application of the theory developed in this work is also discussed, where we have investigated the dynamics of ion solvation in electrolyte solutions by employing the frequency- and wave vector-dependent dynamic response functions.