  We study Yang-Mills connections on holomorphic bundles over complex K\"ahler
manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's
study of flat connections. The space of non-Hermitian Yang-Mills (NHYM)
connections has dimension twice the space of Hermitian Yang-Mills connections,
and is locally isomorphic to the complexification of the space of Hermitian
Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space
of stable bundles). Further, we study the NHYM connections over hyperk\"ahler
manifolds. We construct direct and inverse twistor transform from NHYM bundles
on a hyperk\"ahler manifold to holomorphic bundles over its twistor space. We
study the stability and the modular properties of holomorphic bundles over
twistor spaces, and prove that work of Li and Yau, giving the notion of
stability for bundles over non-K\"ahler manifolds, can be applied to the
twistors. We identify locally the following two spaces: the space of stable
holomorphic bundles on a twistor space of a hyperk\"ahler manifold and the
space of rational curves in the twistor space of the ``Mukai dual''
hyperk\"ahler manifold.
