Abstract: Large networks of nearly identical oscillators, organized as aone-dimensional array or two-dimensional lattice with nonlocal coupling, areable to support peculiar patterns consisting of synchronous and asynchronousregions, called chimera states. Until recently, such chimera states werestudied mainly in the stationary case, when the shape and position of thepattern remain unchanged over time. Inthis talk, I will show that the world of synchronization patterns in nonlocallycoupled systems is much richer. Inparticular, it contains chimera states with periodically modulated macroscopicdynamics, as well as moving chimera states arising from induced or spontaneoussymmetry breaking.  A number ofsurprising properties of such breathing and moving patterns will be reported, including the non-monotonic dependence of thedrift speed on the system parameters, and the secondary spatial modulation of the moving asynchronousregions. Moreover, the constructive roleof small heterogeneity of oscillators for the stability of breathing and moving chimera states will bedemonstrated. From a mathematicalperspective, it will be shown how toperform self-consistency and stability analysis of these chimera states, using the continuum limit formalism, theOtt-Antonsen method and some propertiesof the periodic complex Riccati equation.