  Gauge theories are studied on a space of functions with the Moyal-Weyl
product. The development of these ideas follows the differential geometry of
the usual gauge theories, but several changes are forced upon us. The Leibniz
rule has to be changed such that the theory is now based on a twisted Hopf
algebra. Nevertheless, this twisted symmetry structure leads to conservation
laws. The symmetry has to be extended from Lie algebra valued to enveloping
algebra valued and new vector potentials have to be introduced. As usual, field
equations are subjected to consistency conditions that restrict the possible
models. Some examples are studied.
