  Fix a $C^\infty$ principal $G$--bundle $E^0_G$ on a compact connected Riemann
surface $X$, where $G$ is a connected complex reductive linear algebraic group.
We consider the gradient flow of the Yang--Mills--Higgs functional on the
cotangent bundle of the space of all smooth connections on $E^0_G$. We prove
that this flow preserves the subset of Higgs $G$--bundles, and, furthermore,
the flow emanating from any point of this subset has a limit. Given a Higgs
$G$--bundle, we identify the limit point of the integral curve passing through
it. These generalize the results of the second named author on Higgs vector
bundles.

Differential Geometry