  We describe a unitary matrix model which is constructed from discrete analogs
of the usual projective modules over the noncommutative torus and use it to
construct a lattice version of noncommutative gauge theory. The model is a
discretization of the noncommutative gauge theories that arise from toroidal
compactification of Matrix theory and it includes a recent proposal for a
non-perturbative definition of noncommutative Yang-Mills theory in terms of
twisted reduced models. The model is interpreted as a manifestly star-gauge
invariant lattice formulation of noncommutative gauge theory, which reduces to
ordinary Wilson lattice gauge theory for particular choices of parameters. It
possesses a continuum limit which maintains both finite spacetime volume and
finite noncommutativity scale. We show how the matrix model may be used for
studying the properties of noncommutative gauge theory.
