We summarize our understanding of millisecond radio bursts from an extragalactic population of sources. FRBs occur at an extraordinary rate, thousands per day over the entire sky with radiation energy densities at the source about ten billion times larger than those from Galactic pulsars. We survey FRB phenomenology, source models and host galaxies, coherent radiation models, and the role of plasma propagation effects in burst detection. The FRB field is guaranteed to be exciting: new telescopes will expand the sample from the current ~80 unique burst sources (and a few secure localizations and redshifts) to thousands, with burst localizations that enable host-galaxy redshifts emerging directly from interferometric surveys. * FRBs are now established as an extragalactic phenomenon. * Only a few sources are known to repeat. Despite the failure to redetect other FRBs, they are not inconsistent with all being repeaters. * FRB sources may be new, exotic kinds of objects or known types in extreme circumstances. Many inventive models exist, ranging from alien spacecraft to cosmic strings but those concerning compact objects and supermassive black holes have gained the most attention. A rapidly rotating magnetar is a promising explanation for FRB 121102 along with the persistent source associated with it, but alternative source models are not ruled out for it or other FRBs. * FRBs are powerful tracers of circumsource environments, `missing baryons in the IGM, and dark matter. * The relative contributions of host galaxies and the IGM to propagation effects have yet to be disentangled, so dispersion measure distances have large uncertainties.
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