  The Hitchin-Kobayashi correspondence for vector bundles, established by
Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable
holomorphic vector bundle over a compact Kaehler manifold is stable in the
sense of Takemoto-Mumford if and only if the vector bundle admits a
Hermitian-Einstein metric. Its manifold analogue known as Yau's conjecture,
which originated from Calabi's conjecture, asks whether ``stability'' and
``existence of extremal metrics'' for polarized manifolds are equivalent. In
this note, the recent progress of this subject, by Donaldson, Tian and our
group, together with its relationship to algebraic geometry will be discussed.
