  We develop a theory of stable bundles and affine Hermitian-Einstein metrics
for flat vector bundles over a special affine manifold (a manifold admitting an
atlas whose gluing maps are all locally constant volume-preserving affine
maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the
existence of Hermitian-Einstein metrics on K\"ahler manifolds, and the
extension of this theorem by Li-Yau to the non-K\"ahler complex case of
Gauduchon metrics. Our definition of stability involves only flat vector
subbundles (and not singular subsheaves), and so is simpler than the complex
case in some places.
