  In this paper we give a survey of some models of the integer and fractional
quantum Hall effect based on noncommutative geometry. We begin by recalling
some classical geometry of electrons in solids and the passage to
noncommutative geometry produced by the presence of a magnetic field. We recall
how one can obtain this way a single electron model of the integer quantum Hall
effect. While in the case of the integer quantum Hall effect the underlying
geometry is Euclidean, we then discuss a model of the fractional quantum Hall
effect, which is based on hyperbolic geometry simulating the multi-electron
interactions. We derive the fractional values of the Hall conductance as
integer multiples of orbifold Euler characteristics. We compare the results
with experimental data.
