The structures, electron properties, and potential superconductivity of indium hydrides are systematically studied under high pressure by first-principles density functional calculations. Upon compression, two stable stoichiometries (InH3 and InH5) are predicted to be thermodynamically stable. Particularly, in the two compounds, all hydrogen atoms exist in the form of H-2 or H-3 units. The stable phases present metallic features with the overlap between the conduction and the valence bands. The Bader analysis indicates that charges transfer from In atoms to H atoms. Electron-phonon calculations show that the estimated transition temperatures (T-c) of InH3 and InH5 are 34.1-40.5 and 22.4-27.1 K at 200 and 150 GPa, respectively.