Floquet engineering, the control of quantum systems using periodic driving, is an old concept in condensed matter physics dating back to ideas such as the inverse Faraday effect. However, there is a renewed interest in this concept owing to (a) the rapid developments in laser and ultrafast spectroscopy techniques, (b) discovery and understanding of various “quantum materials” hosting interesting exotic quantum properties, and (c) communication with different areas of physics such as artificial matter and nonequilibrium quantum statistical physics. Here, starting from a nontechnical introduction with emphasis on the Floquet picture and effective Hamiltonians, we review the recent applications of Floquet engineering in ultrafast, nonlinear phenomena in the solid state. In particular, Floquet topological states and their application to ultrafast spintronics and strongly correlated electron systems are overviewed.

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