  Let G be a split semisimple algebraic group with trivial center. Let S be a
compact oriented surface, with or without boundary. We define {\it positive}
representations of the fundamental group of S to G(R), construct explicitly all
positive representations, and prove that they are faithful, discrete, and
positive hyperbolic; the moduli space of positive representations is a
topologically trivial open domain in the space of all representations.
  When S have holes, we defined two moduli spaces closely related to the moduli
spaces of G-local systems on S. We show that they carry a lot of interesting
structures. In particular we define a distinguished collection of coordinate
systems, equivariant under the action of the mapping class group of S. We prove
that their transition functions are subtraction free. Thus we have positive
structures on these moduli spaces. Therefore we can take their points with
values in any positive semifield. Their positive real points provide the two
higher Teichmuller spaces related to G and S, while the points with values in
the tropical semifields provide the lamination spaces. We define the motivic
avatar of the Weil-Petersson form for one of these spaces. It is related to the
motivic dilogarithm.
