The structure and elastic properties of alpha-Zr and its hydrides were investigated by first-principle calculations and experimental methods. Considering all possible H-atom configurations, different phase models of hydrides were constructed. Results show that the stable structures of gamma, delta and epsilon hydrides are P4(2)/mmc, P4(2)/nnm and I4/mmm, respectively. Calculation results suggest that epsilon hydride has the lowest formation enthalpy, and the phase transition sequence of gamma -> delta -> epsilon is proposed. Compared with those of alpha-Zr, the c-axis lattice constants of hydrides become smaller, and the expansion volumes of gamma, delta and epsilon unit cell are 12.1%, 14.8% and 17.9%, respectively. The calculated elastic modulus (E) of the three hydrides are lower than that of alpha-Zr, but their elastic anisotropy is higher than that of alpha-Zr. The elastic properties of alpha-Zr matrix and delta hydride were analyzed by nanoindentation experiment and the results show that E of the alpha-Zr matrix and d hydride is 116.88 and 111.01 GPa, respectively. Therefore, the stress concentration is easy to occur on the hydride sides near the hydrides/matrix interface, so the hydrides are more likely to be the sources of crack and cause brittle fracture of zirconium alloys.