The science of complex systems distinguishes linear from non-linear dynamics. Simpler systems can often be satisfactory described by linear models, but complex systems require non-linear models that can capture more of the characteristics of such systems, such as thresholds, feedback loops, avalanche effects, and irreversibility. Linear systems can be validated by aligning models to the past and using the model to predict the future. Non-linear systems, however, are often time-asymmetric - they can be explained with the wisdom of hindsight, but are not always predictable. For example, systems may respond sharply to minor perturbations, and the quality of this response is a measure of the system resilience. In practice, non-linear dynamics are significant both at the micro-scale of small history and at the macro-scale of deep time. The brilliant young scientist, for example, may experience a series of epiphanies that change his/her understanding and behaviour in an unpredictable and irreversible way. The scientific community as a whole may experience an innovation-cascade that has a similar effect on a much larger scale.Current models of climate change and carbon emission assume the immediate past is a reasonable guide to the future. They struggle to represent the complex causal structures and time-asymmetries of many socio-natural systems. COMPLEX will integrate the quasi-classic models of meso-scale processes with our best understanding of fine-grained space-time patterns and the system-flips that are likely to occur in the long interval between now and 2050. We believe the sub-national region is the key point of entry for studying climate change and its cause-effect interrelations. It is small enough to be sensitive to local factors, large enough to interact with supra-national agencies and stable enough to be historically and culturally distinctive. In addition to undertaking case studies in Norway, Sweden, Netherlands, Spain and Italy, We will develop a suite of modelling tools and decision-support systems to inform national and supra-national policy and support communities across Europe working to make the transition to a low-carbon economy.