Relativistic hydrodynamics has been extensively applied to high energy heavy-ion collisions. We review hydrodynamic calculations for Au+Au collisions at RHIC energies and provide a comprehensive comparison between the model and experimental data. The model provides a very good description of all measured momentum distributions in central and semiperipheral Au+Au collisions, including the momentum anisotropies (elliptic flow) and systematic dependencies on the hadron rest masses up to transverse momenta of about 1.5--2 GeV/c. This provides impressive evidence that the bulk of the fireball matter shows efficient thermalization and behaves hydrodynamically. At higher p_t the hydrodynamic model begins to gradually break down, following an interesting pattern which we discuss. The elliptic flow anisotropy is shown to develop early in the collision and to provide important information about the early expansion stage, pointing to the formation of a highly equilibrated quark-gluon plasma at energy densities well above the deconfinement threshold. Two-particle momentum correlations provide information about the spatial structure of the fireball (size, deformation, flow) at the end of the collision. Hydrodynamic calculations of the two-particle correlation functions do not describe the data very well. Possible origins of the discrepancies are discussed but not fully resolved, and further measurements to help clarify this situation are suggested.