Chimeras are this year coming of age since they were first observed by Kuramoto and Battogtokh in 2002 in a one-dimensional network of complex Ginzburg–Landau equations. What started as an observation of a peculiar coexistence of synchronized and desynchronized states, almost two decades latter turned out to be an important new paradigm of nonlinear dynamics at the interface of physical and life sciences. Chimeras have been observed in uni-hemispheric sleep of aquatic mammals and migratory birds, in electrocorticographic recordings of epileptic seizures, and in neural bump states that are central to the coding of working memory and visual orientation. Chimera states have also been observed experimentally in physical systems, for example in liquid crystal light modulators, and they have been linked to power grids outages and optomechanics.
Here we present a major review of chimeras, dedicated to all aspects of their theoretical and practical existence. We cover different dynamical systems in which chimera states have been observed, different types of chimeras, and different mathematical methods used for their analysis. We also review the importance of network structure for the emergence of chimeras, as well as different schemes aimed at controlling the symmetry breaking spatiotemporal pattern. We conclude by outlining open challenges and opportunities for future research entailing chimeras.