Our understanding of the processes that control the burden and budget of tropospheric ozone has changed dramatically over the last 60 years. Models are the key tools used to understand these changes, and these underscore that there are many processes important in controlling the tropospheric ozone budget. In this critical review, we assess our evolving understanding of these processes, both physical and chemical. We review model simulations from the International Global Atmospheric Chemistry Atmospheric Chemistry and Climate Model Intercomparison Project and Chemistry Climate Modelling Initiative to assess the changes in the tropospheric ozone burden and its budget from 1850 to 2010. Analysis of these data indicates that there has been significant growth in the ozone burden from 1850 to 2000 (approximately 43 + 9%) but smaller growth between 1960 and 2000 (approximately 16 +/- 10%) and that the models simulate burdens of ozone well within recent satellite estimates. The Chemistry Climate Modelling Initiative model ozone budgets indicate that the net chemical production of ozone in the troposphere plateaued in the 1990s and has not changed since then inspite of increases in the burden. ... mehrThere has been a shift in net ozone production in the troposphere being greatest in the northern mid and high latitudes to the northern tropics, driven by the regional evolution of precursor emissions. An analysis of the evolution of tropospheric ozone through the 21st century, as simulated by Climate Model Intercomparison Project Phase 5 models, reveals a large source of uncertainty associated with models themselves (i.e., in the way that they simulate the chemical and physical processes that control tropospheric ozone). This structural uncertainty is greatest in the near term (two to three decades), but emissions scenarios dominate uncertainty in the longer term (2050- 2100) evolution of tropospheric ozone. This intrinsic model uncertainty prevents robust predictions of near-term changes in the tropospheric ozone burden, and we review how progress can be made to reduce this limitation.