Abstract: This is a summary of general theory and new advances in the geometry of discrete groups. Some new results due to the graduate students of the department are mentioned as follows. Let f1 and f2 be two self homeomorphismsof Rn and a discrete convergence group. If f1=f2 in some topological sphere, then f1 =f2 in IP the identity theorem). It is also proved in a very simple way and without applying the Thurston theory, that second kind torsion-free quasiconformal Fuchsian groups of compact type are quasiconformally conjugate to Mobius groups in three dimensions . Some open problems and references are listed .