  We study two-dimensional WZW models with target space a nonreductive Lie
group. Such models exist whenever the Lie group possesses a bi-invariant
metric. We show that such WZW models provide a lagrangian description of the
nonreductive (affine) Sugawara construction. We investigate the gauged WZW
models and we prove that gauging a diagonal subgroup results in a conformal
field theory which can be identified with a coset construction. A large class
of exact four-dimensional string backgrounds arise in this fashion. We then
study the topological conformal field theory resulting from the $G/G$ coset. We
identify the Kazama algebra extending the BRST algebra, and the BV algebra
structure in BRST cohomology which it induces.
