  The low--energy excited states of a system of interacting one--dimensional
fermions in a conducting state are collective charge and spin density
oscillations. The unusual physical properties of such a system (called
``Luttinger liquid'') are characterized by the velocities $u_\rho$ and
$u_\sigma$ of the charge and spin excitations, as well as by a parameter
$K_\rho$ that determines the power law behavior of correlation functions.
Umklapp scattering occuring at half--filling or other commensurate bandfilling
can lead to a transition into an insulating state, characterized in particular
by a gap in the charge excitations (the Mott--Hubbard gap). The properties in
the vicinity of the transition are shown to depend on both the way the
transition is approached (constant bandfilling and varying interaction, or
constant interaction and varying bandfilling) and on the ``order'' of the
commensurability. In particular, even and odd fractional fillings show quite
different behavior. This behavior is illustrated in detail using lattice models
like the Hubbard model and its extensions.
