  In this thesis, the emerging field of higher gauge theory will be discussed,
particularly in relation to problems arising in M-theory, such as selfdual
strings and the so-called (2,0) theory. This thesis will begin with a Nahm-like
construction for selfdual strings using loop space, the space of loops on
spacetime. This construction maps solutions of the Basu-Harvey equation, the
BPS equation arising in the description of multiple M2-branes, to solutions of
a selfdual string equation on loop space. Furthermore, all ingredients of the
construction reduce to those of the ordinary Nahm construction when
compactified on a circle with all loops restricted to those wrapping the
circle. The rest of this thesis, however, will not involve loop space. We will
see a Nahm-like construction for the case of infinitely many selfdual strings,
suspended between two M5-branes. This is possible since the limit taken renders
the fields describing the M5-branes abelian. This avoids the problem which the
rest of this thesis focuses on: What fields describe multiple M5-branes? The
answer is likely to involve higher gauge theory, a categorification of gauge
theory which describes the parallel transport of extended objects. Any theories
which involves 3-algebras, including current M2-brane models and the
Lambert-Papageorgakis M5-brane model, are examples of higher gauge theories.
Recently, a class of models with N=(1,0) supersymmetry have been found, with
significant overlap with algebraic structures in higher gauge theory. This
overlap suggests that the full N=(2,0) theory could involve semistrict
$L_\infty$-algebras. Finally, we will see some explicit selfdual string
solutions, which may fit into these frameworks.
