It is found that historical simulations successfully reproduce the spatio-temporal structure of the CPT temperature in comparison to reanalysis data. The interannual variability associated with El Niño-Southern Oscillation and intraseasonal variability associated with equatorial waves are also reasonably well captured. However, the models show non-negligible biases in several aspects: 1) most models have a warm bias around the CPT; 2) large inter-model differences exist in the amplitude of the seasonal cycle in upper-tropospheric and lower-stratospheric temperature; 3) many models overestimate lower stratospheric warming in response to volcanic aerosols; 4) temperature variability associated with the quasi-biennial oscillation and Madden-Julian oscillation is absent in most models; 5) equatorial waves near the CPT exhibit a wide range of variations among the models with unrealistically persistent Kelvin waves in several models. In the RCP 8.5 scenario, the models predict robust warming at 100 hPa and cooling at 70 hPa. A weakened seasonal cycle in the temperatures of both the 100- and 70-hPa levels is also predicted in most models. These findings may have important implications for cross-tropopause water vapor transport and related global climate change and variability.