Powdered samples of the perovskite BaSnO3 exhibit strong near-infrared (NIR) luminescence at room temperature, following band-gap excitation at 380 nm (3.26 eV). The emission spectrum is characterized by a broad band centered at 905 nm (1.4 eV), tailing on the high-energy side to approximately 760 nm. The Stokes shift is 1.9 eV, and measured lifetimes in the range 7-18 ms depend on preparative conditions. These extraordinary long values indicate that the luminescence involves a defect state(s). At low temperatures, both a sharp peak and a broad band appear in the visible portion of the luminescence spectrum at approximately 595 nm. Upon cooling, the intensity of the NIR emission decreases, while the integrated intensities of the visible emission features increase to approximately 40% of the NIR intensity at 77 K. Room-temperature photoluminescence (PL) is observed across the Ba1-xSrxSnO3 series. As the strontium content increases, the excitation maximum and band gap shift further into the UV, while the intensity of the NIR emission peak decreases and shifts further into the infrared. This combination leads to an unexpectedly large increase in the Stokes shift. The unusual NIR PL in BaSnO3 may originate from recombination of a photogenerated valence-band hole and an occupied donor level, probably associated with a Sn2+ ion situated roughly 1.4 eV above the valence-band edge.