  We prove an analogue of the Hitchin-Kobayashi correspondence for compact,
oriented, taut Riemannian foliated manifolds with transverse Hermitian
structure. In particular, our Hitchin-Kobayashi theorem holds on any compact
Sasakian manifold. We define the notion of stability for foliated Hermitian
vector bundles with transverse holomorphic structure and prove that such
bundles admit a basic Hermitian-Einstein connection if and only if they are
polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to
the foliated setting. We relate the transverse Hermitian-Einstein equations to
higher dimensional instanton equations and in particular we look at the
relation to higher contact instantons on Sasaki manifolds. For foliations of
complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We
also demonstrate that the weak Uhlenbeck compactness theorem fails in general
for basic connections on a foliated bundle. This shows that not every result in
gauge theory carries over to the foliated setting.
