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We study nontrivial (i.e., containing more than one orbit) minimal sets of the geodesic flow on Γ/T1\Bbb H^2, where Γ is a nonelementary Fuchsian group. It is not difficult to prove that nontrivial compact minimal sets always exist. We establish the existence of nontrivial noncompact minimal sets in two cases: (1) Γ is a Schottky group of special kind generated by infinitely many hyperbolic elements, (2) Γ contains a parabolic element (in particular, Γ = PSL(2, \Bbb Z)). This is done by geometric coding of geodesic orbits and constructing a minimal set for symbolic dynamics with infinite alphabet.

Let \Lambda \subset \partial {\Bbb H}^{2} be the limit set of a Fuchsian group \Gamma \subset \operatorname{Iso} {\Bbb H}^{2}. The behavior of geodesic and horocycle trajectories on the unit tangent bundle \Gamma \backslash T^{1} {\Bbb H}^{2} allows one to introduce different subclasses of \Lambda such as the conical limit set \Lambda_{c} \subset \Lambda, the horocyclic limit set \Lambda_{h} \subset \Lambda , etc. We consider a class of infinitely generated Schottky groups obtained by placing countably many disjoint semicircles in {\Bbb H}^{2} with a single accumulation point in \partial {\Bbb H}^{2}, and pairing them by hyperbolic transformations. For all such groups we give a complete description of \Lambda_{c} \subset \partial {\Bbb H}^{2} and of all other subclasses of \Lambda related to the dynamics of the geodesic flow on \Gamma \backslash T^{1} {\Bbb H}^{2}. This is given in terms of natural combinatorial coding of limit points and is the same for all groups \Gamma of this form. In contrast, we show that the structure of horocycle orbits is very sensitive to geometric parameters of the semicircles nearby the accumulation point. In particular, we construct an infinitely generated Schottky group such that the horocycle flow on T^{1} {\Bbb H}^{2} has only two types of orbits: those dense in the nonwandering subset \Omega^{+} \subset T^{1} {\Bbb H}^{2} and those divergent in both directions. This shows that infinitely generated Fuchsian group need not have Garnett limit points.

Here we survey the results on the structure of Fuchsian groups due to Hopf, Hedlund, Sullivan, Nicholls, Pommerenke, and others from the viewpoint of the dynamics of the geodesic and horocycle flows on the corresponding surfaces. Special attention is given to the structure of horocycle orbits; in particular, we construct Fuchsian groups with new types of horocycle orbits which are neither closed nor dense in the nonwandering set. We give a unique classification of Fuchsian groups from the dynamical viewpoint and indicate some open problems.