  We propose a field theoretical model defined on non-commutative space-time
with non-constant non-commutativity parameter $\Theta(x)$, which satisfies two
main requirements: it is gauge invariant and reproduces in the commutative
limit, $\Theta\to 0$, the standard $U(1)$ gauge theory. We work in the slowly
varying field approximation where higher derivatives terms in the star
commutator are neglected and the latter is approximated by the Poisson bracket,
$-i[f,g]_\star\approx\{f,g\}$. We derive an explicit expression for both the NC
deformation of Abelian gauge transformations which close the algebra
$[\delta_f,\delta_g]A=\delta_{\{f,g\}}A$, and the NC field strength ${\cal F}$,
covariant under these transformations, $\delta_f {\cal F}=\{{\cal F},f\}$. NC
Chern-Simons equations are equivalent to the requirement that the NC field
strength, ${\cal F}$, should vanish identically. Such equations are
non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the
gauge invariant action, $S=\int {\cal F}^2$. As guiding example, the case of
$su(2)$-like non-commutativity, corresponding to rotationally invariant NC
space, is worked out in detail.
