  Non-commutative gauge theories with a non-constant NC-parameter are
investigated. As a novel approach, we propose that such theories should admit
an underlying L$_\infty$ algebra, that governs not only the action of the
symmetries but also the dynamics of the theory. Our approach is well motivated
from string theory. We recall that such field theories arise in the context of
branes in WZW models and briefly comment on its appearance for integrable
deformations of AdS$_5$ sigma models. For the SU(2) WZW model, we show that the
earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be
bootstrapped via an L$_\infty$ algebra. We then apply this approach to the
construction of non-commutative Chern-Simons and Yang-Mills theories on flat
and curved backgrounds with non-constant NC-structure. More concretely, up to
the second order, we demonstrate how derivative and curvature corrections to
the equations of motion can be bootstrapped in an algebraic way from the
L$_\infty$ algebra. The appearance of a non-trivial A$_\infty$ algebra is
discussed, as well.
