  We calculate quantum averages of Wilson loops (holonomies) in gauge theories
on the Euclidean noncommutative plane, using a path-integral representation of
the star-product. We show how the perturbative expansion emerges from a concise
general formula and demonstrate its anomalous behavior at large parameter of
noncommutativity for the simplest nonplanar diagram of genus 1. We discuss
various UV/IR regularizations of the two-dimensional noncommutative gauge
theory in the axial gauge and, using the noncommutative loop equation,
construct a consistent regularization.
