The anionic transport in single crystals with a fluorite-type structure of the nominally pure BaF2, the dilute doped BaF2 with 0.01 and 0.1 mole % LaF3 and the anion-excess binary Ba1-xLaxF2+x (0 less than or equal to x less than or equal to 0.5), Ba0.85R0.15F2.15 (R = La, Sm, and Lu), and Ba0.8R0.2F2.2 (R = La, Ce, Nd, Gd, Tb, and Ho) solid solutions were studied by impedance spectroscopy in the temperature range from ambient to 1073 K. At high temperatures the conductivity plots for the concentrated solid solutions display a considerable deviation from Arrhenius behaviour. This is caused by the diffuse Faraday phase transition in these materials. The microscopic origin of the anionic conductivity depends mainly on the nature of the defect structure in the solid solutions. By controlling the composition of Ba1-xRxF2+x crystals the fluoride ionic conductivity can reach a value of about 1 S cm(-1) at 1073 K.