  The fractional quantum Hall effect, being one of the most studied phenomena
in condensed matter physics during the past thirty years, has generated many
groundbreaking new ideas and concepts. Very early on it was realized that the
zoo of emerging states of matter would need to be understood in a systematic
manner. The first attempts to do this, by Haldane and Halperin, set an agenda
for further work which has continued to this day. Since that time the idea of
hierarchies of quasiparticles condensing to form new states has been a pillar
of our understanding of fractional quantum Hall physics. In the thirty years
that have passed since then, a number of new directions of thought have
advanced our understanding of fractional quantum Hall states, and have extended
it in new and unexpected ways. Among these directions is the extensive use of
topological quantum field theories and conformal field theories, the
application of the ideas of composite bosons and fermions, and the study of
nonabelian quantum Hall liquids. This article aims to present a comprehensive
overview of this field, including the most recent developments.
