This thesis surveys parts of the forthcoming joint work [23] in which the non-abelianization map of [16] was extended from the case of G = SL(n) and G = GL(n) to the case of arbitrary reductive algebraic groups. The non-abelianization map is an algebraic map from a moduli space of certain N -local systems on the complement of a divisor P in a punctured Riemann surface X, to the moduli space of G-local systems on X.