Studies of the X-ray luminescence and scintillation properties of a SrMoO 4 crystal as function of temperature down to T=10 K have been carried out. The luminescence in SrMoO 4 is quenched at room temperature, but below T<200 K the crystal exhibits a broad emission band with a maximum at a wavelength of 520 nm. The emission is attributed to the radiative decay of self-trapped excitons and defects acting as traps for the exactions at low temperatures. Such complex character of radiative decay is reflected in the kinetics which contains several components plus a contribution from delayed recombination at low temperatures. The temperature dependence of scintillation light output of SrMoO 4 was studied. Comparing with a reference ZnWO 4 crystal measured under the same experimental conditions it was found that the light output of SrMoO 4 is 15±5%. It is suggested, therefore, that there is scope for optimisation of strontium molybdate for application as scintillator in cryogenic rare event searches.