Fast radio bursts (FRBs) are millisecond-time-scale bursts of coherent radio emission that are luminous enough to be detectable at cosmological distances. In this Review, I describe the discovery of FRBs, subsequent advances in understanding them, and future prospects. Thousands of potentially observable FRBs reach Earth every day, which likely originate from highly magnetic and/or rapidly rotating neutron stars in the distant Universe. Some FRBs repeat, with this subclass often occurring in highly magnetic environments. Two repeating FRBs exhibit cyclic activity windows, consistent with an orbital period. One nearby FRB was emitted by a Galactic magnetar during an x-ray outburst. The host galaxies of some FRBs have been located, providing information about the host environments and the total baryonic content of the Universe.
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