  We review the notions of (weak) Hermitian-Yang-Mills structure and
approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we
construct the Donaldson functional for Higgs bundles over compact K\"ahler
manifolds and we present some basic properties of it. In particular, we show
that its gradient flow can be written in terms of the mean curvature of the
Hitchin-Simpson connection. We also study some properties of the evolution
equation associated to that functional. Next, we study the problem of the
existence of approximate Hermitian-Yang-Mills structure and its relation with
the algebro-geometric notion of Mumford-Takemoto semistability and we show that
for a Higgs bundle over a compact Riemann surface, the notion of approximate
Hermitian-Yang- Mills structure is in fact the differential-geometric
counterpart of the notion of semistability. Finally, using Donaldson heat flow,
we show that the semistability of a Higgs bundle over a compact K\"ahler
manifold (of every dimension) implies the existence of an approximate
Hermitian-Yang-Mills structure. As a consequence of this we deduce that many
results about Higgs bundles written in terms of approximate
Hermitian-Yang-Mills structures can be translated in terms of semistability.
