We consider representations of the fundamental group of an orientable surface into a real or complex Lie group and study some of the algebraic, geometric and quantization properties of the spaces of isomorphism classes of such representations.

In particular, we focus on their relation to some deformation and moduli spaces in geometry: the Teichmüller and Schottky spaces of marked Riemann surfaces, and the moduli spaces of semistable vector and Higgs bundles over a Riemann surface, presenting some classical and some more recent results.

We also describe methods of geometric quantization of Kähler manifolds and show that, using a coherent state transform, one obtains a satisfactory quantization of some of these spaces, providing a unified treatment of abelian and genus 1 non-abelian theta functions.