SrMgF4 has an extremely large bandgap Eg of 12.50 eV as obtained from reflection dispersion. The symmetry of this crystal is monoclinic P21 at room temperature and transforms to the orthorhombic Cmc21 phase near 478 K as the temperature increases. The acentric character of the low-temperature (LT) phase is confirmed by pyroelectric luminescence at T < 440 K. The fundamental absorption edge of the LT phase is located at 122 nm (10.15 eV). A considerable difference between the absorption edge and bandgap Eg is due to the strong exciton absorption. The first-principles electronic structure, refractive indices, nonlinear susceptibility and polarizability were calculated for both LT and high-temperature (HT) phases. Band-to-band transitions are direct for the LT phase but indirect for HT. In spite of relatively low birefringence (∼0.017) and nonlinear susceptibility (∼0.044 pm V(-1), an order lower than that in KDP), SrMgF4 crystals are considered promising for nonlinear optics thanks to their transparency far in the vacuum ultraviolet spectral region.