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We review how the international modelling community, encompassing Integrated Assessment models, global and regional Earth system and climate models, and impact models, have worked together over the past few decades, to advance understanding of Earth system change and its impacts on society and the environment, and support international climate policy. We then recommend a number of priority research areas for the coming ~6 years (i.e. until ~2030), a timescale that matches a number of newly starting international modelling activities and encompasses the IPCC 7th Assessment Report (AR7) and the 2nd UNFCCC Global Stocktake. Progress in these areas will significantly advance our understanding of Earth system change and its impacts and increase the quality and utility of science support to climate policy. We emphasize the need for continued improvement in our understanding of, and ability to simulate, the coupled Earth system and the impacts of Earth system change. There is an urgent need to investigate plausible pathways and emission scenarios that realize the Paris Climate Targets, including pathways that overshoot the 1.5 °C and 2 °C targets, before later returning to them. Earth System models (ESMs) need to be capable of thoroughly assessing such warming overshoots, in particular, the efficacy of negative CO2 emission actions in reducing atmospheric CO2 and driving global cooling. An improved assessment of the long-term consequences of stabilizing climate at 1.5 °C or 2 °C above pre-industrial temperatures is also required. We recommend ESMs run overshoot scenarios in CO2-emission mode, to more fully represent coupled climate - carbon cycle feedbacks. Regional downscaling and impact models should also use forcing data from these simulations, so impact and regional climate projections are as realistic as possible. An accurate simulation of the observed record remains a key requirement of models, as does accurate simulation of key metrics, such as the Effective Climate Sensitivity. For adaptation, improved guidance on potential changes in climate extremes and the modes of variability these extremes develop in, is a key demand. Such improvements will most likely be realized through a combination of increased model resolution and improvement of key parameterizations. We propose a deeper collaboration across modelling efforts targeting increased process realism and coupling, enhanced model resolution, parameterization improvement, and data-driven Machine Learning methods. With respect to sampling future uncertainty, increased collaboration between approaches that emphasize large model ensembles and those focussed on statistical emulation is required. We recommend increased attention is paid to High Impact Low Likelihood (HILL) outcomes. In particular, the risk and consequences of exceeding critical tipping points during a warming overshoot. For a comprehensive assessment of the impacts of Earth system change, including impacts arising directly from specific mitigation actions, it is important detailed, disaggregated information from the Integrated Assessment Models (IAMs) used to generate future scenarios is available to impact models. Conversely, methods need to be developed to incorporate potential future societal responses to the impacts of Earth system change into scenario development. Finally, the new models, simulations, data, and scientific advances, proposed in this article will not be possible without long-term development and maintenance of a robust, globally connected infrastructure ecosystem. This system must be easily accessible and useable across all modelling communities and across the world, allowing the global research community to be fully engaged in developing and delivering new scientific knowledge to support international climate policy. [ABSTRACT FROM AUTHOR]