X-ray diffraction of Ce(3+)-doped SrMgF(4) (SMF:Ce) crystals shows a superlattice structure, reflecting the distribution of Ce(3+) polyhedra centres observed in optical experiments. Optical absorption bands and fluorescence bands from the Ce(3+) polyhedra centres overlap in the vacuum ultraviolet (VUV) and ultraviolet (UV) regions, respectively, so that wide pumping and tuning ranges are expected for laser operation. The SMF:Ce crystals, as well as the isomorphous BaMgF(4), are candidates for a tunable laser gain material with nonlinear properties. The optical absorption, excitation, and fluorescence bands observed in the SMF:Ce crystals at low temperatures are ascribed to five distinct fluorescent centres. Three centres have well-known Ce(3+) optical characters, for example, fluorescence with double peaks separated by 2000 cm(-1) and five resolved absorption/excitation bands. These centres are assigned to Ce(3+)-polyhedra classified by weak and strong crystal fields as a consequence of the superlattice structure. The other two fluorescence bands observed in the visible region have 1.5-2 times larger linewidths than those of the former three bands. These bands are interpreted as optical transitions from complexes consisting of Ce(3+) and one or two electrons trapped at a vacancy of the nearest neighbour F(-) ligand ions.