  We study the perturbative approach to the Wilsonian integration of
noncommutative gauge theories in the matrix representation. We begin by
motivating the study of noncommutative gauge theories and reviewing the matrix
formulation. We then systematically develop the perturbative treatment of UV
states and calculate both the leading and next to leading order one- and
two-loop corrections to the quantum effective action. Throughout, we discuss
how our formalism clarifies problems associated with UV-IR mixing, a particular
emphasis being placed on the dipole structure imposed by noncommutative gauge
invariance. Ultimately, using the structural understanding developed in this
work, we are able to determine the exact form of perturbative corrections in
the UV regime defined by $\theta\Lambda^2\gg 1$. Finally, we apply our results
to the analysis of the divergence structure and show that 3+1 and higher
dimensional noncommutative theories that allow renormalization beyond one-loop
are not self-consistent.
