  The physics of the fractional quantum Hall effect is the physics of
interacting electrons confined to a macroscopically degenerate Landau level. In
this Chapter we discuss the theory of the quantum Hall effect in systems where
the electrons have degrees of freedom in addition to the two-dimensional
orbital degree of freedom. We will be primarily interested in the situation
where a finite number of states, most-commonly two, are available for each
orbital state within a degenerate Landau level and will refer to these systems
as multi-component systems. Physical realizations of the additional degree of
freedom include the electron spin, the valley index in multi-valley
semiconductors, and the layer index in multiple-quantum-well systems. The
consideration of multi-component systems expands the taxonomy of incompressible
states and fractionally charged excitations, and for example, leads to the
appearance of fractions with even denominators. More interestingly, it also
leads us to new physics, including novel spontaneously broken symmetries and in
some cases, finite temperature phase transitions. We present an introduction to
this rich subject.
