For a semisimple real Lie group $G$, parabolic $G$-Higgs bundles over a punctured Riemann surface emerge as the appropriate holomorphic objects corresponding to fundamental group representations of the surface with fixed holonomy around these punctures. In the absense of punctures, the study of the moduli spaces of (non-parabolic) $G$-Higgs bundles is a highly non-trivial and particularly fruitful subject, that has come to be the rapidly growing area known nowadays as Higher Teichmüller Theory. In this talk, we shall survey on results recently obtained in the course of establishing a Higher Teichmüller Theory for parabolic $G$-Higgs moduli. Joint work with Hao Sun and Lutian Zhao.