In this talk we will cover the basics of quantization of moduli spaces of bundles over Riemann Surfaces. This will include both moduli spaces for compact Lie groups and also for non compact ones, which can also be though of as moduli spaces of Higgs bundles. In particular we will discuss quantisation with respect to both complex and real polarisations, the old and new Verlinde formulae and construct Hitchin-Witten connections so as to obtain quantum representation of mapping class groups and further their relations to conformal field theory and TQFT.