  We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact
Hermitian manifolds. This correspondence refers to moduli spaces of "universal
holomorphic oriented pairs". Most of the classical moduli problems in complex
geometry (e. g. holomorphic bundles with reductive structure groups,
holomorphic pairs, holomorphic Higgs pairs, Witten triples, arbitrary quiver
moduli problems) are special cases of this universal classification problem.
Our Kobayashi-Hitchin correspondence relates the complex geometric concept
"polystable oriented holomorphic pair" to the existence of a reduction solving
a generalized Hermitian-Einstein equation. The proof is based on the
Uhlenbeck-Yau continuity method.
  We also investigate metric properties of the moduli spaces in this general
non-Kaehlerian framework. We discuss in more detail moduli spaces of oriented
connections, Douady Quot spaces and moduli spaces of non-abelian monopoles.
