  We study the competition between a random potential and a commensurate
potential on interacting fermionic and bosonic systems using a variety of
methods. We focus on one dimensional interacting fermionic systems but higher
dimensional bosonic and fermionic extensions, as well as classical equivalents
are also discussed. Our methods which include bosonization, replica variational
method, functional renormalization group (RG) and perturbation around the
atomic limit, go beyond conventional perturbative expansions around the
Luttinger liquid in one dimension. All these methods agree on the prediction in
these systems of a phase, the Mott glass, intermediate between the Anderson
Insulator (compressible, with a pseudogap in the optical conductivity) and the
Mott Insulator (incompressible with a gap in the optical conductivity). The
Mott glass, which was unexpected from a perturbative renormalization group
point of view has a pseudogap in the conductivity while remaining
incompressible. Having derived the existence of the Mott Glass phase in one
dimension, we show qualitatively that its existence can also be expected in
higher dimension. We discuss the relevance of this phase to experimental
systems such as disordered classical elastic systems and dirty bosons.
