The Lagrangian view of air flow in the middle atmosphere is studied by computing tens of thousands of particle trajectories in a global primitive-equation model. This approach focuses attention on the long-term fate of air and provides a high-resolution view of air transport. Simulations are carried out for both the northern and southern winters to permit an inter-hemispheric comparison. The first part of the paper concentrates on transport in the meridional plane. Trajectories followed for many months show patterns which are well organized on a global scale. Ensembles of particles followed for a much shorter time are used to calculate a Lagrangian mean meridional circulation which shows close similarities to published diabatic circulations calculated from observational data. Maps of meridional dispersion reveal a broad mid-latitude band, with well defined boundaries, in which rapid dispersion occurs. A more detailed visualization of transport in and around the southern polar vortex in late winter shows that small scales are generated in this mid-latitude region, but not in the vortex nor at low latitudes. The ability of the vortex to retain a labelled air mass for many months is clearly demonstrated, and the relevance of the results to understanding springtime ozone depletion is discussed.