Heavy metal fluoride glasses of varying alkali metal fluorides have been studied by positron annihilation lifetime spectroscopy. The alkali series 53ZrF4·(40-x)BaF2·4LaF3·3AlF3·xLiF is compared with the mixed alkali series 53ZrF4·20BaF2·4LaF3·(20-x)CsF·xLiF. The mean lifetime and refractive index are non-linear functions of the dopant concentration, (40-x)BaF2·xLiF, in the alkali series as compared with the linear dependence of the mean lifetime and refractive index on the dopant concentration, (20-x)CsF·xLiF, found in the mixed alkali series. The alkali series spectra can be best fitted by a single lifetime, while the mixed alkali series spectra can be best fitted with two lifetime components. Probable states of the positron at annihilation in these glasses are discussed and compared with those states found in oxide glasses. There is no evidence for ortho-positronium annihilation in these fluoride glasses.