TITLE: Moduli spaces of connections as representation spaces
QUESTION [9 upvotes]: 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 $G$. 

Are there other moduli spaces which can be identified with representation spaces?

REPLY [5 votes]: Another interesting example, not mentioned yet, where character varieties come up is the moduli space of polygons.
Moduli spaces of $(G,X)$-structures on compact manifolds are also related to character varieties via the Ehresmann--Thurston Theorem.  See here for example.
They are also related to spaces of phylogenetic trees and spaces of spin networks.