  We consider a BRST approach to G/H coset WZNW models, {\it i.e.} a
formulation in which the coset is defined by a BRST condition. We will give the
precise ingrediences needed for this formulation. Then we will prove the
equivalence of this approach to the conventional coset formulation by solving
the the BRST cohomology. This will reveal a remarkable connection between
integrable representations and a class of non-integrable representations for
negative levels. The latter representations are also connected to string
theories based on non-compact WZNW models. The partition functions of G/H
cosets are also considered. The BRST approach enables a covariant construction
of these, which does not rely on the decomposition of G as $G/H\times H$. We
show that for the well-studied examples of $SU(2)_k \times SU(2)_1/SU(2)_{k+1}$
and $SU(2)_k/U(1)$, we exactly reproduce the previously known results.
