For a non-compact reductive Lie group $G$, the notion of $G$-Higgs bundles over a compact Riemann surface $X$, of genus $g \geq2$, was introduced by Hitchin (80s and 90s). These are appropriate objects for extending the non-abelian Hodge Theorem (the work of Corlette, Donaldson, Hitchin and Simpson) to representations of the fundamental group in a real reductive Lie group $G$. Motivated partially by this identification, the moduli space of $G$-Higgs bundles has been extensively studied. Here we give the obstructions to a deformation retraction from the moduli spaces of $G$-Higgs bundles to the moduli space of semistable principal bundles over $X$, in contrast with the situation when $g = 1$. The existence of those obstructions allows us to deduce the reducibility of the nilpotent cone of the moduli space of $G$-Higgs bundles: that is the pre-image of zero under the Hitchin map. All concepts will be motivated with several examples, and we will give an overview of known results on the moduli spaces of G-Higgs bundles as well as some open problems.