A formalism for the application of variational transition-state theory and semiclassical vibrationally adiabatic transmission coefficients to bimolecular reactions involving an arbitrary number of atoms is presented. This generalizes previous work on atom-diatom reactions. We make applications in this paper to the reactions OH+H2→H2O+H and OH+D 2→HDO+D using the Schatz-Elgersma fit to the Walch-Dunning ab initio potential energy surface. For both reactions we find large differences between conventional and variational transition-state theory and large effects of anharmonicity on the calculated rate constants. The effect of reaction-path curvature on the calculated transmission coefficients and rate constants is also large. The final calculated values of the kinetic isotope effects are in good agreement with experiment at high temperature but too large at room temperature.