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The emerging field of complex light{#x2014}the study and application of custom light beams with tailored intensity, polarization or phase{#x2014}is a focal point for fundamental breakthroughs in optical science. As this review will show, those advances in fundamental understanding, coupled with the latest developments in complex light generation, are translating into a range of diverse and cross-disciplinary applications that span microscopy, high-data-rate communications, optical trapping and quantum optics. We can expect more twists along the way, too, as researchers seek to manipulate and control the propagation speed of complex light beams, while others push the more exotic possibilities afforded by complex light in quantum-entanglement experiments.