Surveys of the GHz radio sky with sub-millisecond time resolution have revealed a population of astrophysical fast radio bursts (FRBs), lasting up to a few milliseconds and with peak flux densities rivaling the brightest quasars. Their cold-plasma dispersion delays, and levels of pulse-broadening due to multi-path propagation, are significantly in excess of expectations from the Milky Way along their sightlines. They are thus identified as coherent outbursts from extragalactic compact objects. This, however, remains the extent of our knowledge. We do not know from which objects in which galaxies, and at what distances, they originate. I will discuss possible insights into these basic properties from analyses of observed FRBs, and also provide a status update on an FRB localization machine being built at Caltech's Owens Valley radio observatory. In addition to representing a new, likely exotic form of radio source, FRBs provide the opportunity for tomographic studies of the density, turbulence, and magnetization of plasma in the interstellar media of distant galaxies, in galactic halos, and in the intergalactic medium.