  We review the remarkable properties, including superconductivity,
charge-density-wave ordering, and metal-insulator transitions, of lead- and
potassium-doped barium bismuthate. We discuss some of the early theoretical
studies of these systems. Our recent theoretical work, on the negative-$U\/$,
extended-Hubbard model for these systems, is also described. Both the large-
and intermediate-$U\/$ regimes of this model are examined, using mean-field and
random-phase approximations, particularly with a view to fitting various
experimental properties of these bismuthates. On the basis of our studies, we
point out possibilities for exotic physics in these systems. We also emphasize
the different consequences of electronic and phonon-mediated mechanisms for the
negative $U.\/$ We show that, for an electronic mechanism, the \secin
\,\,phases of these bismuthates must be unique, with their transport properties
{\it dominated by charge $\pm 2e$ Cooperon bound states}. This can explain the
observed difference between the optical and transport gaps. We propose other
experimental tests for this novel mechanism of charge transport and comment on
the effects of disorder.
