A study of the temperature and polarization dependence of the absorption and luminescence from electron and neutron-irradiated Mg${\mathrm{F}}_{2}$ crystals has been made. The absorption band at 260 nm is accepted as due to an $F$ center, and excitation, when other centers are absent, produces an emission at about 430 nm. Two other absorption bands at 370 and 400 nm arise from different types of $M$ centers, and our data suggest that these centers give rise to emission bands at 420 and 600 nm, respectively. Several narrow absorption lines are observed in both electron-and neutron-irradiated samples. From polarization studies and optical bleaching treatment, the lines at 387.2, 385.7, 384.5, 382.3, and 380.8 nm are identified as zero-phonon and phonon-assisted transitions of the defect responsible for the 370-nm band. A comparison is made between the vibrational energies associated with the phonon-assisted lines and the dispersion relationships calculated for Mg${\mathrm{F}}_{2}$.