  Gauge theory on the q-deformed two-dimensional Euclidean plane R^2_q is
studied using two different approaches. We first formulate the theory using the
natural algebraic structures on R^2_q, such as a covariant differential
calculus, a frame of one-forms and invariant integration. We then consider a
suitable star product, and introduce a natural way to implement the
Seiberg-Witten map. In both approaches, gauge invariance requires a suitable
``measure'' in the action, breaking the E_q(2)-invariance. Some possibilities
to avoid this conclusion using additional terms in the action are proposed.
