Accurate quantum reactive scattering time-dependent wave packet close-coupling calculations have been carried out to determine total reaction probabilities and integral cross-sections for the O++ H2 → OH++H reaction in a range of collision energies from 10-3 eV up to 1.0 eV for the H2 rovibrational states (v = 0; j = 0, 1, 2) and (v = 1; j = 0) using the potential energy surface (PES) by Mart ́ınez. As expected for a barrierless reaction, the reaction cross section decays rapidly with collision energy, Ec, following a behavior that nearly corresponds to that predicted by the Langevin model. Rotational excitation of H2 into j=1,2 has a very moderate effect on reactivity, similarly to what happens with vibrational excitation below Ec ≈0.3 eV. However, at higher collision energies the cross section increases notably when H2 is promoted to v=1. This effect is explained by resort- ing to the effective potentials in the entrance channel. The integral cross sections have been used to calculate rate constants in the temperature range 200-1000 K. A good overall agreement has been found with the available experimental data on integral cross sections and rate constants. In addition, time-independent quantum mechanical and quasi-classical trajectory (QCT) calculations have been performed on the same PES aimed to compare the various methodologies and to discern the detailed mechanism of the title reaction. In particular, the analysis of individual trajectories has made possible to explain, in terms of the coupling between reagent relative velocity and the topography of the PES, the presence of a series of alternating maxima and minima in the collision energy dependence of the QCT reaction probabilities for the reactions with H2(v = 0, 1, j = 0), which are absent in the quantum mechanical calculations.