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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 …