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The role played by hydrogen impurities in the conductivity of indium oxide (In2O3) has been controversial. Some studies, based on the effect of oxygen partial pressure in growth or annealing environments, argue that oxygen vacancies are the cause of the conductivity of In2O3. However, there is a growing body of theoretical and experimental work which suggests that hydrogen centers can be important shallow donors in In2O3. Muon-spin-resonance experiments indicate that implanted muons, whose properties mimic those of hydrogen, form shallow donors in In2O3. In2O3 thin films containing hydrogen show n-type conductivity with high mobility, and theory states that interstitial hydrogen (Hi+) and hydrogen trapped at an oxygen vacancy (HO+) are shallow donors that can give rise to n-type conductivity or compensate acceptors in In2O3. We have performed a series of IR absorption experiments and complementary theory to determine the properties of OH and OD centers in In2O3 single crystals. Annealing In2O3 samples in H2 or D2 at temperatures near 450oC produces an n-type layer≈0.06 mm thick with an n-type doping of 1.6×1019 cm-3. The resulting free-carrier absorption is correlated with an OH center with a vibrational frequency of 3306 cm-1 that we associate with interstitial H+, which is thermally stable around 600oC. Additional O-H (O-D) vibrational lines are assigned to metastable configurations of the interstitial H+ (D+) center and complexes of H (D) with In vacancies. Unlike other oxides studied recently where H trapped at an oxygen vacancy is the dominant shallow donor (ZnO and SnO2, for example), interstitial H+ is found to be the dominant H-related shallow donor in In2O3. In addition, the diffusion of hydrogen defects has been studied by thinning experiments and isothermal anneals.