The problem of reconstructing nonlinear and complex dynamical systems from measured data or time series is central to many scientific disciplines including physical, biological, computer, and social sciences, as well as engineering and economics. In this paper, we review the recent advances in this forefront and rapidly evolving field, aiming to cover topics such as compressive sensing (a novel optimization paradigm for sparse-signal reconstruction), noised-induced dynamical mapping, perturbations, reverse engineering, synchronization, inner composition alignment, global silencing, Granger Causality and alternative optimization algorithms. Often, these rely on various concepts from statistical and nonlinear physics such as phase transitions, bifurcation, stabilities, and robustness. The methodologies have the potential to significantly improve our ability to understand a variety of complex dynamical systems ranging from gene regulatory systems to social networks towards the ultimate goal of controlling such systems. Despite recent progress, many challenges remain. A purpose of this Review is then to point out the specific difficulties as they arise from different contexts, so as to stimulate further efforts in this interdisciplinary field.
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