Magnesium-based alloys attract significant interest as cost-efficient hydrogen storage materials allowing the combination of high gravimetric storage capacity of hydrogen with fast rates of hydrogen uptake and release and pronounced destabilization of the metal–hydrogen bonding in comparison with binary Mg–H systems. In this review, various groups of magnesium compounds are considered, including (1) RE–Mg–Ni hydrides (RE = La, Pr, Nd); (2) Mg alloys with p-elements (X = Si, Ge, Sn, and Al); and (3) magnesium alloys with d-elements (Ti, Fe, Co, Ni, Cu, Zn, Pd). The hydrogenation–disproportionation–desorption–recombination process in the Mg-based alloys (LaMg12, LaMg11Ni) and unusually high-pressure hydrides synthesized at pressures exceeding 100 MPa (MgNi2H3) and stabilized by Ni–H bonding are also discussed. The paper reviews interrelations between the properties of the Mg-based hydrides and p–T conditions of the metal–hydrogen interactions, chemical composition of the initial alloys, their crystal structures, and microstructural state.