  The first part of this paper is a review of the author's work with S. Bahcall
which gave an elementary derivation of the Chern Simons description of the
Quantum Hall effect for filling fraction $1/n$. The notation has been
modernized to conform with standard gauge theory conventions.
  In the second part arguments are given to support the claim that abelian
non-commutative Chern Simons theory at level $n$ is exactly equivalent to the
Laughlin theory at filling fraction $1/n$. The theory may also be formulated as
a matrix theory similar to that describing D0-branes in string theory. Finally
it can also be thought of as the quantum theory of mappings between two
non-commutative spaces, the first being the target space and the second being
the base space.
