We present a lattice formulation of non-commutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of non-commutative field theories is demonstrated at a completely non-perturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and non-commutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic non-commutative gauge theories in the continuum can be regularized non perturbatively by means of ordinary lattice gauge theory with 't Hooft flux. In the case of irrational non-commutativity 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 non-commutative Yang-Mills theories using twisted large-N reduced models. We study the coupling of non-commutative 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 non-commutative 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 non-commutative quantum electrodynamics.