The study of discrete subgroups of Lie groups has started in the works of F.Klein and H.Poincare in the 19th century, and has experineced enormous growth thanks to A.Selberg, A.Borel, G.Mostow, G.Margulis and many others in the 20th century. The study of discrete subgroups of Diff(M^1), however, is a very new subject where one often tries to push the theory parallel to the theory of discrete subgroups of Lie groups. In this talk, I'll mainly concentrate on two related questions: existence of discrete free subgroups and (a weaker version of) Margulis Lemma. The calssical Margulis Lemma states that a connected Lie group G posseses an open neighborhood U of identity such that any discrete subgroup of G generated by elements from U is nilpotent.