Membrane technologies are widely demanded in a number of modern industries. Ion-exchange membranes are one of the most widespread and demanded types of membranes. Their main task is the selective transfer of certain ions and prevention of transfer of other ions or molecules, and the most important characteristics are ionic conductivity and selectivity of transfer processes. Both parameters are determined by ionic and molecular mobility in membranes. To study this mobility, the main techniques used are nuclear magnetic resonance and impedance spectroscopy. In this comprehensive review, mechanisms of transfer processes in various ion-exchange membranes, including homogeneous, heterogeneous, and hybrid ones, are discussed. Correlations of structures of ion-exchange membranes and their hydration with ion transport mechanisms are also reviewed. The features of proton transfer, which plays a decisive role in the membrane used in fuel cells and electrolyzers, are highlighted. These devices largely determine development of hydrogen energy in the modern world. The features of ion transfer in heterogeneous and hybrid membranes with inorganic nanoparticles are also discussed.