  We extend Hitchin's results on "The self-duality equations on a Riemann
surface" (Proc. LMS (3), vol. 55, 1987) to orbifold Riemann surfaces. We prove
existence results for orbifold solutions of the Yang-Mills-Higgs equations and
construct the moduli space of solutions. These moduli spaces provide
interesting examples of non-compact hyper-Kahler manifolds in all dimensions
divisible by 4 and of completely integrable Hamiltonian systems. We also
reinterpret these moduli spaces as spaces of orbifold Higgs bundles and as
representation varieties of Fuchsian groups.
