We examine the sensitivity of the Arctic Oscillation (AO) index to increases in greenhouse gas concentrations in integrations of five climate models (the Hadley Centre coupled models (HadCM2 and HadCM3), the European Centre/Hamburg models (ECHAM3 and ECHAM4), and the Goddard Institute for Space Studies stratosphere-resolving (GISS-S) model) and in the National Centers for Environmental Prediction reanalysis. With the exception of HadCM2 all the models show a significant positive AO response to greenhouse gas forcing, but in the models lacking a well-resolved stratosphere that response is smaller than observed. In these models the AO index is linearly dependent on the radiative forcing, even up to ~20 times current CO2 levels. By contrast, the GISS-S stratosphere-resolving model shows an AO response comparable to that observed, but the sensitivity of the model to further increases in forcing is reduced when CO2 levels exceed ~1.5 times preindustrial values. It has been suggested that greenhouse gas forcing results in the equatorward deflection of planetary waves, which leads to a cooling and strengthening of the polar vortex and hence an increase in the surface Arctic Oscillation. In the observations the number of sudden warmings has reduced dramatically, consistent with this planetary wave effect, leading to a large mean cooling of the vortex. However, neither the GISS-S nor the HadCM3 models are able to reproduce the observed temperature changes, suggesting that this explanation for the impact of the inclusion of a stratosphere in the model may be incomplete.