The existence of extragalactic fast radio bursts (FRBs) of sub-millisecond durations, originating at cosmologically significant distances, has been established. A population of repeating FRB sources is emerging, and three burst sources have been identified with host galaxies at redshifts between 0.19 and 0.66. Explaining the FRB phenomenon has proved a compelling challenge to theory, with the number of distinct models only this year being superseded by the ~80 reported events. I will review the state of the field, with a focus on recent results that are beginning to unravel the FRB mystery. Additionally, FRBs have opened a powerful new window into otherwise unseen matter in the Universe. I will show how large FRB samples will help assess the baryon contents and physical conditions in the hot/diffuse circumgalactic, intracluster, and intergalactic medium.