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ABSTRACT: A “complex system” is in general any system comprised of a great number of heterogeneous entities, among which local interactions create multiple levels of collective structure and organization. Examples include natural systems ranging from bio-molecules and living cells to human social systems and the ecosphere, as well as sophisticated artificial systems such as the Internet, power grid or any large-scale distributed software system. A unique feature of complex systems, generally overlooked by traditional science, is the emergence of non-trivial superstructures which often dominate the system's behaviour and cannot be easily traced back to the properties of the constituent entities. Not only do higher emergent features of complex systems arise from lower-level interactions, but the global patterns that they create affect in turn these lower levels — a feedback loop sometimes called immergence. In many cases, complex systems possess striking properties of robustness against various large-scale and potentially disruptive perturbations. They have an inherent capacity to adapt and maintain their stability. Because complex systems require analysis at many different spatial and temporal scales, scientists face radically new challenges when trying to observe complex systems, in learning how to describe them effectively, and in developing original theories of their behaviour and control. The aim of this roadmap is to identify a set of wide thematic domains for complex systems research over the next five years. Each domain is organized around a specific question or topic and proposes a relevant set of “grand challenges”, i.e., clearly identifiable problems whose solution would stimulate significant progress in both theoretical methods and experimental strategies.