The structures of five phases in the Ba1-xKxBiO3 system for 0x0.5 and temperatures below 473 K have been determined by neutron powder diffraction. Bulk superconductivity occurs only in a cubic perovskite phase which exists for x0.37 (at 10 K). At room temperature, as the potassium concentration is decreased, the cubic structure distorts first by BiO6 octahedral tilting and then by symmetric oxygen breathing-mode distortions. Semiconducting behavior for the monoclinic phase at 0x0.1 can readily be explained on the basis of a commensurate charge-density wave. The semiconducting orthorhombic phase, which extends from the monoclinic phase to the cubic superconducting phase, contains only octahedral tilting distortions that would not be expected to destroy metallic behavior. Thus, this commensurate structure provides no explanation for the metal-insulator transition. An incommensurate modulation extending across the semiconducting region of the phase diagram has been observed by electron diffraction, but is not observed by neutron diffraction. It is not clear whether this incommensurate modulation is responsible for the nonmetallic behavior. © 1990 The American Physical Society.