  We give a review of truncated L$_\infty$ algebras, as used in the study of
higher gauge theory. These structures are believed to hold the correct
properties to adequately describe gauge theory of extended objects. We discuss
how to construct topological higher-gauge-invariant theories and how their
solutions relate to multisymplectic geometries. We also show how Courant
algebroids fit into this formalism, so as to be able to study higher gauge
theory on generalized geometric bundles, i.e. on $ T\Sigma \oplus T^*\Sigma $,
for some space-time $\Sigma $. We will see that via this formalism we can match
and explain a recently proposed M5-brane model, arrived at in a more heuristic
way, whose field content seemed difficult to interpret but finds a natural
motivation in this framework.
