The Brewer-Dobson circulation (BDC) is a global-scale atmospheric circulation which is characterized by rising tropical tropospheric air into the stratosphere, poleward movement, and descending air in the middle and high latitudes. The BDC controls the transport of air and trace gases in the stratosphere and mesosphere and is of high relevance for ozone distribution. This thesis gives an overview on the historical and theoretical background of the BDC and on the current status of knowledge from observations and models. The concept of the BDC tries to explain why tropical stratospheric air has less ozone than polar air, even though most ozone is produced in the tropical stratosphere. The BDC consists of two branches. A lower branch controls the transport of air in the tropical lower stratosphere and an upper branch is effective in the upper stratosphere and mesosphere. The mass transport by the BDC includes two components: The zonal mean residual circulation and two-way quasi-horizontal mixing. Both processes are driven by atmospheric waves on different spatial and temporal scales. In winter, planetary-scale tropospheric Rossby waves and, in all seasons, gravity waves propagate upward, dissipate, transfer their momentum, and induce a meridional circulation. Increased wave breaking in the extratropics strengthens it.Information on the BDC and its characteristic is mainly derived from atmospheric models, reanalyses, and observations of tracers. In a changing climate due to increasing greenhouse gas concentrations model simulations indicate an acceleration of the BDC. Information from observations is partly inconsistent due to the sparseness and uncertainty of measurements.Changes in the generation, propagation, and dissipation of waves are considered mainly responsible for BDC changes but specific causes for the BDC trends are still under debate.The effect of a changing BDC for stratosphere-troposphere exchange and for weather and climate is a topic of active research.