The Brewer-Dobson circulation (BDC) is a key feature of the stratosphere that models need to accurately represent in order to simulate surface climate variability and change adequately. For the first time, the Climate Model Intercomparison Project includes in its phase 6 (CMIP6) a set of diagnostics that allow for careful evaluation of the BDC. Here, the BDC is evaluated against observations and reanalyses using historical simulations. CMIP6 results confirm the well-known inconsistency in the sign of BDC trends between observations and models in the middle and upper stratosphere. Nevertheless, the large uncertainty in the observational trend estimates opens the door to compatibility. In particular, when accounting for the limited sampling of the observations, model and observational trend error bars overlap in 40 % of the simulations with available output. The increasing CO.sub.2 simulations feature an acceleration of the BDC but reveal a large spread in the middle-to-upper stratospheric trends, possibly related to the parameterized gravity wave forcing. The very close connection between the shallow branch of the residual circulation and surface temperature is highlighted, which is absent in the deep branch. The trends in mean age of air are shown to be more robust throughout the stratosphere than those in the residual circulation.