The stable isotope composition of biogenic apatite is an important geochemical marker that can record environmental parameters and is widely used to infer past climates, biomineralization processes, dietary preferences and habitat of vertebrates. In this study, theoretical equilibrium isotopic fractionation of oxygen, carbon and calcium in hydroxyapatite and carbonate-bearing hydroxyapatite is investigated using first-principles methods based on density-functional theory and compared to the theoretical isotopic fractionation properties of calcite, CO 2 and H 2 O. Considering the variability of apatite crystal-chemistry, special attention is given to specific contributions of crystal sites to isotopic fractionation. Significant internal fractionation is calculated for oxygen and carbon isotopes in CO 3 between the different structural sites occupied by carbonate groups in apatite (typically 7‰ for both 18 O/ 16 O and 13 C/ 12 C fractionation at 37 °C). Compared with calcite-water oxygen isotope fractionation, occurrence of A-type substitution in apatite structure, in addition to the main B-type substitution, could explain the larger temperature dependence of oxygen isotope fractionation measured at low temperature between carbonate in apatite and water. Theoretical internal fractionation of oxygen isotopes between carbonate and phosphate in B-type carbonated apatite ($8‰ at 37 °C) is consistent with experimental values obtained from modern and well-preserved fossil bio-apatites. Concerning calcium, theoretical results suggest a small fractionation between apatite and calcite (À0.17‰ at 37 °C). Internal fractionation reaching 0.8‰ at 37 °C occurs between the two Ca sites in hydroxyapatite. Furthermore, the Ca isotopic fractionation properties of apatite are affected by the occurrence of carbonate groups, which could contribute to the variability observed on natural samples. Owing to the complexity of apatite crystal-chemistry and in light of the theoretical results, measurements of site-specific isotopic fractionation properties could improve our understanding and the interpretation of isotopic records in apatites.