  We present non-perturbative results for U(1) gauge theory in spaces, which
include a non-commutative plane. In contrast to the commutative space, such
gauge theories involve a Yang-Mills term, and the Wilson loop is complex on the
non-perturbative level. We first consider the 2d case: small Wilson loops
follows an area law, whereas for large Wilson loops the complex phase rises
linearly with the area. In four dimensions the behavior is qualitatively
similar for loops in the non-commutative plane, whereas the loops in other
planes follow closely the commutative pattern. In d=2 our results can be
extrapolated safely to the continuum limit, and in d=4 we report on recent
progress towards this goal.
