It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi\_1(M), G) / G$. 


Furthermore, Atiyah and Bott (1983) showed that over a closed surface the moduli space of (central) Yang-Mills connections can be seen as the space of representations of a group $\Gamma\_R$ (which is a central extension of the fundamental group by $\mathbb{R}$) to $G$. 



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Moduli spaces of connections as representation spaces
Narasimhan and Seshadri proved that the moduli space of semi-stable vector bundles of rank $r$ on a compact Riemann surface is canonically isomorphic to the moduli space of unitary representations of degree $r$. This has been extended to projective manifolds by Donaldson, then to compact Kähler manifolds by Uhlenbeck-Yau; and to the case of principal $G$-bundles (for $G$ complex reductive) and representations of a maximal compact subgroup by Ramanathan and Subramanian.


