  We review recent results on the derivation of a global path integral density
for Yang-Mills theory. Based on a generalization of the stochastic quantization
scheme and its geometrical interpretation we first recall how locally a
modified Faddeev-Popov path integral density for the quantization of Yang-Mills
theory can be derived, the modification consisting in the presence of specific
finite contributions of the pure gauge degrees of freedom. Due to the Gribov
problem the gauge fixing can be defined only locally and the whole space of
gauge potentials has to be partitioned into patches. We discuss a global
extension of the path integral by summing over all patches, which can be proven
to be manifestly independent of the specific local choices of patches and gauge
fixing conditions, respectively. In addition to the formulation on the whole
space of gauge potentials we discuss the corresponding path integral on the
gauge orbit space.
