  Let F be a finitely generated discrete group. Given a covering map H to G of
Lie groups with G either compact or complex reductive, there is an induced
covering map Hom(F, H) to Hom(F, G). We show that when the fundamental group of
G is torsion-free and F is free, free Abelian, or the fundamental group of a
closed Riemann surface M of genus g, this map induces a covering map between
the corresponding moduli spaces of representations. We give conditions under
which this map is actually the universal covering, leading to new information
regarding fundamental groups of these moduli spaces. Let pi be the fundamental
group of M. As an application, we show that for g>0, the stable moduli space
Hom(pi, SU)/SU is homotopy equivalent to infinite complex projective space.
  In the Appendix by Ho and Liu, it is shown show that there is a bijection
between the number of connected components of Hom(pi, G) and the fundamental
group of [G,G] for all complex connected reductive Lie groups G.
