| Sumario: | Each group G of automorphisms of a compact Riemann surface of genus bigger than one can be written as the (finite) quotient G = Γ/Λ for some Fuchsian groups Γ and Λ where Λ is torsion-free. Amongst the natural questions which arise in this situation, this survey article focuses on the following ones. First, which groups can be realized as the full group of all automorphisms of some compact Riemann surface? Second, does every Fuchsian group Γ contain a torsion-free normal subgroup Λ of finite index? Third, for a fixed finite group G, is it possible to characterize all Fuchsian groups Γ containing a torsion-free normal subgroup Λ such that G = Γ/Λ? And fourth, which conditions assure that a group G of automorphisms of a compact Riemann surface is the full group of all its automorphisms? We will consider these topics not only in the setting of compact Riemann surfaces and Fuchsian groups but also in the setting of compact Klein surfaces and non-euclidean crystallographic groups.
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