This book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It covers the fundamental problems of two-dimensional quantum statistics in terms of topology and the corresponding braid group formalism for composite fermions, and the main formalism used in many-body quantum Hall theories, the Chern-Simons theory. Numerical studies are introduced for spherical systems and the composite fermion theory is tested. The book introduces the concept of the heirarchy of condensed states, the BCS paired Hall state, and multi-component quantum Hall systems and spin quantum Hall systems.