This article reviews the experimental and theoretical investigations of the superconducting solid solutions BaPb$_{1-x}$Bi$_x$O$_3$ (BPB), which have critical temperatures $T_c\approx$ 13 K. The crystal structure, structural phase transitions, and the electrical and optical properties are examined in detail. Methods of preparing ceramic samples, as well as thin films and single crystals, are discussed briefly. Measurements of the electron specific heat and of the upper critical magnetic field are interpreted in terms of a theory of superconductors with partial dielectrization of the electron spectrum. Particular attention is paid to those properties of BPB which are a consequence of the granularity of the ceramic macrostructure and the existence of weak Josephson links between the granules. Correlations between the composition dependences of various normal and superconducting characteristics of BPB are elucidated, and the nature of the superconducting high Tc state with a small electron density of states ($N$(0) = 3$\cdot$10$^{21}$ eV$^{-1}$ cm$^{-3}$) at the Fermi surface is discussed.