  We present a lattice formulation of noncommutative Yang-Mills theory in
arbitrary even dimensionality. The UV/IR mixing characteristic of
noncommutative field theories is demonstrated at a completely nonperturbative
level. We prove a discrete Morita equivalence between ordinary Yang-Mills
theory with multi-valued gauge fields and noncommutative Yang-Mills theory with
periodic gauge fields. Using this equivalence, we show that generic
noncommutative gauge theories in the continuum can be regularized
nonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooft
flux. In the case of irrational noncommutativity parameters, the rank of the
gauge group of the commutative lattice theory must be sent to infinity in the
continuum limit. As a special case, the construction includes the recent
description of noncommutative Yang-Mills theories using twisted large $N$
reduced models. We study the coupling of noncommutative gauge fields to matter
fields in the fundamental representation of the gauge group using the lattice
formalism. The large mass expansion is used to describe the physical meaning of
Wilson loops in noncommutative gauge theories. We also demonstrate Morita
equivalence in the presence of fundamental matter fields and use this property
to comment on the calculation of the beta-function in noncommutative quantum
electrodynamics.
