We describe the main differences in simulations of stratospheric climate and variability by models within the fifth Coupled Model Intercomparison Project (CMIP5) that have a model top above the stratopause and relatively fine stratospheric vertical resolution (high‐top), and those that have a model top below the stratopause (low‐top). Although the simulation of mean stratospheric climate by the two model ensembles is similar, the low‐top model ensemble has very weak stratospheric variability on daily and interannual time scales. The frequency of major sudden stratospheric warming events is strongly underestimated by the low‐top models with less than half the frequency of events observed in the reanalysis data and high‐top models. The lack of stratospheric variability in the low‐top models affects their stratosphere‐troposphere coupling, resulting in short‐lived anomalies in the Northern Annular Mode, which do not produce long‐lasting tropospheric impacts, as seen in observations. The lack of stratospheric variability, however, does not appear to have any impact on the ability of the low‐top models to reproduce past stratospheric temperature trends. We find little improvement in the simulation of decadal variability for the high‐top models compared to the low‐top, which is likely related to the fact that neither ensemble produces a realistic dynamical response to volcanic eruptions. Keypoints We assess and compare the performance of CMIP5 models in the stratosphere.Low‐top models lack stratospheric variability.Stratosphere‐troposphere coupling is hence weaker in low‐top models.