  We carry an intrinsic approach to the study of the connectedness of the
moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann
surface, when $G$ is a complex reductive (not necessarily connected) Lie group.
We prove that the number of connected components of $\mathcal{M}_G$ is indexed
by the corresponding topological invariants. In particular, this gives an
alternative proof of the counting by J. Li of the number of connected
components of the moduli space of flat $G$-connections in the case in which $G$
is connected and semisimple.
