To every transformation of an uncountable set one can naturally associate a group of transformations which is "full" in some sense. The notion of full group was defined by H.A. Dye in 1959. These groups turned out to be of extreme importance in various areas of dynamical systems, especially in the theory of orbit equivalence in measurable, Borel, and Cantor dynamics. I will discuss some old and new results about full groups, their algebraic and topological properties. The talk (possibly continued the following week) is supposed to be self-contained; all necessary definitions will be given.