  We define a new homotopy algebraic structure, that we call a braided
$L_\infty$-algebra, and use it to systematically construct a new class of
noncommutative field theories, that we call braided field theories. Braided
field theories have gauge symmetries which realize a braided Lie algebra, whose
Noether identities are inhomogeneous extensions of the classical identities,
and which do not act on the solutions of the field equations. We use Drinfel'd
twist deformation quantization techniques to generate new noncommutative
deformations of classical field theories with braided gauge symmetries, which
we compare to the more conventional theories with star-gauge symmetries. We
apply our formalism to introduce a braided version of general relativity
without matter fields in the Einstein-Cartan-Palatini formalism. In the limit
of vanishing deformation parameter, the braided theory of noncommutative
gravity reduces to classical gravity without any extensions.
