  We discuss the quantum mechanics of a particle in a magnetic field when its
position x^{\mu} is restricted to a periodic lattice, while its momentum
p^{\mu} is restricted to a periodic dual lattice. Through these considerations
we define non-commutative geometry on the lattice. This leads to a deformation
of the algebra of functions on the lattice, such that their product involves a
``diamond'' product, which becomes the star product in the continuum limit. We
apply these results to construct non-commutative U(1) and U(M) gauge theories,
and show that they are equivalent to a pure U(NM) matrix theory, where N^{2} is
the number of lattice points.
