  Reducing the many-fermion problem to a set of single-particle (s.p.)
equations, the Kohn-Sham (KS) theory has provided a practical tool to implement
\textit{ab initio} calculations of ground-state energies and densities in
many-electron systems. There have been attempts to extend the KS theory so that
it could describe other physical quantities, or it could be applied to other
many-fermion systems. By generalizing and reformulating the KS theory in terms
of the 1-body density matrix, we expose the minimal composition of the theory
that enables the reduction of the many-fermion problem to the s.p. equations.
Based on the reformulation, several basic issues are reconsidered. The $v$- and
$N$-representabilities for the KS theory are distinguished from those for the
Hohenberg-Kohn theorem. Criteria for the extendability of the KS theory are
addressed.
