Variational transition state theory with adiabatic and least-action ground-state transmission coefficients is applied to calculate reaction rates for O(3P)+H2→OH+H in both collinear and three-dimensional worlds for temperatures of 200-1400 K. Five different potential energy surfaces are considered. The collinear studies are used to assess the accuracy of the dynamical and energetic approximations, which include the no-recrossing assumption of generalized transition state theory, semiclassical methods for tunneling calculations, and a Morse approximation for quantizing the generalized-transition-state stretching vibrations. Although the potential energy surfaces show wide differences in behavior, the calculations with least-action ground-state transmission coefficients agree with the accurate quantal results within a factor of 2.8 in all cases. We find that the three-dimensional reaction is dominated by tunneling at room temperature and nearby for all five surfaces. For the calculations on the most accurate ab initio potential energy surface, 60% of the ground-state reaction proceeds by tunneling even at 400 K. The tunneling fractions rise dramatically as the temperature is lowered. The calculations show that quantitative estimation of the tunneling effects requires consideration of reaction-path curvature and of tunneling paths that deviate from the minimum energy path.