Frisch et al. have estimated, by comparing ab initio potential energy and dynamics calculations to experiment, that the potential energy barrier for the title reaction is 2.7 kcal/mol, which is considerably higher than the 0.7-0.9 kcal/mol barriers on recent semiempirical surfaces we have constructed. In the present article we report a global potential energy surface with a 2.7 kcal/mol barrier and stretch and bend potentials based on ab initio calculations. Variational transition state theory and cross section calculations, based on this surface and including anharmonicity and multidimensional tunneling, are in very poor agreement with experiment, e.g., the thermal rate constant at 190 K is 30 times lower than the experimental value for F+H2 and 35 times lower than the experimental value for F+D2. We also report additional model calculations that pinpoint the reasons for the difference between the results of Frisch et al. and ours. The two main reasons are (i) Frisch et al. approximate the vibrationally adiabatic barrier height by its value at the saddle point rather than by its higher variational value, and (ii) Frisch et al. neglect important anharmonic effects in the bending degrees of freedom. This analysis identifies those features of the potential energy surface which it would be most useful to address in any future ab initio calculations directed to the threshold behavior of the F+H2 reaction or its isotopic analogs.