Let X be a compact Riemann surface and G be a compact Lie group. The space of flat G connections on X modulo gauge transformations appears in (at least) two other ways:

• as the quotient under conjugation of the space of G-representations of the fundamental group of X;
• As the moduli space of semistable holomorphic vector bundles on X (with some additional data).

We will describe these spaces together with explicit equivalences between them, and some of their geometric and topological properties. Finally we present some recent results related to the geometric quantization of these spaces in real and in Kähler polarizations.