  We give a pedagogical introduction to $L_\infty$-algebras and their uses in
organising the symmetries and dynamics of classical field theories, as well as
of the conventional noncommutative gauge theories that arise as low-energy
effective field theories in string theory. We review recent developments which
formulate field theories with braided gauge symmetries as a new means of
overcoming several obstacles in the standard noncommutative theories, such as
the restrictions on gauge algebras and matter fields. These theories can be
constructed by using techniques from Drinfel'd twist deformation theory, which
we review in some detail, and their symmetries and dynamics are controlled by a
new homotopy algebraic structure called a 'braided $L_\infty$-algebra'. We
expand and elaborate on several novel theoretical issues surrounding these
constructions, and present three new explicit examples: the standard
noncommutative scalar field theory (regarded as a braided field theory), a
braided version of $BF$ theory in arbitrary dimensions (regarded as a higher
gauge theory), and a new braided version of noncommutative Yang-Mills theory
for arbitrary gauge algebras.
