The Brewer-Dobson circulation (BDC) is a key feature of the stratosphere that models need to accurately represent in order to improve the representation of surface climate variability. 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 BDC trends between observations and models in the middle and upper stratosphere. The increasing CO2 simulations feature a robust acceleration of the BDC but also reveal large uncertainties in the deep branch trends. The very close connection between the shallow branch 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. The paper reflects the current knowledge and main uncertainties regarding the BDC.