  In this paper, the key ideas of characterizing universality classes of
dissipation-free (incompressible) quantum Hall fluids by mathematical objects
called quantum Hall lattices are reviewed. Many general theorems about the
classification of quantum Hall lattices are stated and their physical
implications are discussed. Physically relevant subclasses of quantum Hall
lattices are defined and completely classified. The results are carefully
compared with experimental data and also with other theoretical schemes (the
hierarchy schemes). Several proposals for new experiments are made which could
help to settle interesting issues in the theory of the (fractional) quantum
Hall effect and thus would lead to a deeper understanding of this remarkable
effect.
