We have determined the crystal structures of five phases in the Bal-xKxBiO3 system for 0≤x≤0.5 and 10K≤T≤473K. Superconductivity occurs only in a cubic perovskite phase with the maximum Tc at the transition to an orthorhombic, Ibmm, phase. The commensurate structure of this orthorhombic phase, which involves only tilting of the Bi06 octahedra, provides no explanation for its non-metallic behavior. However, an incommensurate structural modulation, observed by electron diffraction, may provide a rationalization. Thus, the maximum Tc at the phase boundary may result from competition between superconductivity and a charge density wave. Further studies of the phase diagram at high temperatures, where controlled oxygen atmospheres are required, reveal an unusual reentrant formation of a potassium-containing impurity phase which explains the need for synthesis techniques that involve initially reacting in a reducing atmosphere, followed by oxygenation at lower temperature.