  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 solutions of
the evolution equation associated with that functional. Next, we study the
problem of the existence of approximate Hermitian-Yang-Mills structures and its
relation with the algebro-geometric notion of semistability and we show that
for 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, we review the notion of admissible hermitian structure
on a torsion-free Higgs sheaf and define the Donaldson functional for such an
object.
