  This is a survey article whose main goal is to explain how many components of
the character variety of a closed surface are either deformation spaces of
representations into the maximal compact subgroup or deformation spaces of
certain Fuchsian representations. This latter family is of particular interest
and is related to the field of higher Teichm\"uller theory. Our main tool is
the theory of Higgs bundles. We try to develop the general theory of Higgs
bundles for real groups and indicate where subtleties arise. However, the main
emphasis is placed on concrete examples which are our motivating objects. In
particular, we do not prove any of the foundational theorems, rather we state
them and show how they can be used to prove interesting statements about
components of the character variety. We have also not spent any time developing
the tools (harmonic maps) which define the bridge between Higgs bundles and the
character variety. For this side of the story we refer the reader to the survey
article of Q. Li [arXiv:1809.05747].
