  The non-perturbative validity of covariant BRST-quantization of gauge
theories on compact Euclidean space-time manifolds is reviewed.
BRST-quantization is related to the construction of a Topological Quantum Field
Theory (TQFT) of Witten type on the gauge group. The criterion for the
non-perturbative validity of the quantization is that the partition function of
the corresponding TQFT does not vanish and that its (equivariant) BRST-algebra
is free of anomalies. I sketch the construction of a TQFT whose partition
function is proportional to the generalized Euler-characteristic of the coset
space $SU(n)_{gauge}/SU(n)_{global}$ with an associated equivariant
BRST-algebra that manifestly preserves translational symmetry. Some
non-perturbative consequences of this approach are discussed.
