The effects of pressure on the crystal structure of the three known polymorphs of magnesium sulfate (α-MgSO4, β-MgSO4, and γ-MgSO4) have been theoretically studied by means of density-functional theory calculations up to 45 GPa. We determined that under ambient conditions γ-MgSO4 is an unstable polymorph, which decomposes into MgO + SO3, and that the response of the other two polymorphs to hydrostatic pressure is non-isotropic. Additionally, we found that at all pressures β-MgSO4 has a larger enthalpy than α-MgSO4. This indicates that β-MgSO4 is thermodynamically unstable versus α-MgSO4 and predicts the occurrence of a β-α phase transition under moderate compression. Our calculations also predict the existence under pressure of additional phase transitions to two new polymorphs of MgSO4, which we named δ-MgSO4 and ε-MgSO4. The α-δ transition is predicted to occur at 17.5 GPa, and the δ-ε transition at 35 GPa, pressures that nowadays can be experimentally easily achieved. All the predicted structural transformations are characterized as first-order transitions. This suggests that they can be non-reversible, and therefore the new polymorphs could be recovered as metastable polymorphs under ambient conditions. The crystal structure of the two new polymorphs is reported. In them, the coordination number of sulfur is four as in the previously known polymorphs, but the coordination number of magnesium is eight instead of six. In this article we will report the axial and bond compressibility for the four polymorphs of MgSO4. The pressure-volume equation of state of each phase is also given, which is described by a third-order Birch-Murnaghan equation. The values obtained for the bulk modulus are 62 GPa, 57 GPa, 102 GPa, and 119 GPa for α-MgSO4, β-MgSO4, δ-MgSO4, and ε-MgSO4, respectively. Finally, the electronic band structure of these four polymorphs of MgSO4 has been calculated for the first time. The obtained results will be presented and discussed.