Quantum reactive scattering calculations are reported for the four-atom reactions OH + H2(upsilon = 0,1)-->H2O + H, OH + D2-->HOD + D, and OD + H2-->DOH + H, and their reverse reactions. The method involves using hyperspherical coordinates to describe the H2 vibration and one local OH stretching vibration of H2O, accounting for both the rotation of the OH and the bending mode of H2O with a spherical harmonic basis set, and applying a version of the bending-corrected rotating-line approximation to treat the rotation of H2 and the vibration of initial OH. The method gives cross sections and rate coefficients for these reactions which are state selected in the initial OH (j) rotational and H2(upsilon) vibrational states and in the H2O(n,m) product states where n and m label bending and local OH-stretching vibrational states of H2O. A modified potential-energy surface based on a fit to ab initio data is used. The calculated rate coefficients for both the OH + H2(upsilon = 0) and OH + D2(upsilon = 0) reactions agree very well with experiment over the whole temperature range of 250-1050 K. The rate coefficients for OH + H2(upsilon = 0) are significantly larger than those for OH + D2(upsilon = 0) at lower temperatures due to quantum tunneling. The rate coefficient for the OH + H2(upsilon = 1) reaction at room temperature also agrees well with experiment, and is larger than that for the OH + H2(upsilon = 0) reaction by factors ranging from 10(3) at 150 K to 10 at 1000 K. The rate coefficients for the OD + H2(upsilon = 0) reaction are almost identical to those for the OH + H2(upsilon = 0) reaction. The rotational product distributions of OH(j) arising from the reverse reaction H2O + H-->OH(j) + H2 are found to be correlated with excitation in the initial vibrational bending mode of H2O.