Epsomite is known to undergo a series of phase transitions up to 3 GPa at room temperature, a pressure regime which is of relevance to the interiors of icy planetary bodies, where MgSO4-hydrates may be important 'rock-forming' minerals. In collaboration with colleagues at the Institute for High Pressure Physics (Troitsk, Russia) we have just completed a study of protonated and deuterated epsomite in the range 0 < P < 2.8 GPa, 77 < T < 295 K using a piston-cylinder device. These studies have provided us with values for the density of epsomite, as well as the bulk- and shear-moduli as a function of P and T. The new work has shown the need for new high-pressure crystallographic measurements, which are the basis of this proposal.