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

Studying Deformations of Fuchsian Representations with Higgs Bundles