Projections of future atmospheric composition change and its impacts on air quality and climate depend heavily on chemistry-climate models that allow us to investigate the effects of changing emissions and meteorology. These models are imperfect as they rely on our understanding of the chemical, physical and dynamical processes governing atmospheric composition, on the approximations needed to represent these numerically, and on the limitations of the observations required to constrain them. Model intercomparison studies show substantial diversity in results that reflect underlying uncertainties, but little progress has been made in explaining the causes of this or in identifying the weaknesses in process understanding or representation that could lead to improved models and to better scientific understanding. Global sensitivity analysis provides a valuable method of identifying and quantifying the main causes of diversity in current models. For the first time, we apply Gaussian process emulation with three independent global chemistry transport models to quantify the sensitivity of ozone and hydroxyl radicals (OH) to important climate-relevant variables, poorly-characterized processes and uncertain emissions. We show a clear sensitivity of tropospheric ozone to atmospheric humidity and precursor emissions which is similar for the models, but find large differences between models for methane lifetime, highlighting substantial differences in the sensitivity of OH to primary and secondary production. This approach allows us to identify key areas where model improvements are required while providing valuable new insight into the processes driving tropospheric composition change.