This thesis is based on two papers by the author and consists of two parts.
We review the recent developments in the theory of multiple M2-branes and
3-algebras leading to multiple D2-brane theories. The inclusion of flux terms
for the supersymmetric BLG and ABJM theories of closed M2-branes is discussed
and then generalised to open M2-branes. Here the boundary condition is derived
and different BPS configurations are examined where we find a mass deformed
Basu-Harvey equation for the M2-M5 system. The Lorentzian 3-algebra is then
employed for obtaining a theory of D2-branes in a flux background, we then
obtain the new fuzzy funnel solution of the system of D2-D4 branes in a flux.
Matrix theories and their compactifications as well as noncommutative geometry
and noncommutative gauge theories are reviewed with a discussion of their
generalisations to three dimensions to be used to describe the M-theory three
form potential C3. A new feature of string theory is then obtained called
the quantum Nambu geometry (QNG). It is found by considering the action for
D1-strings in a RR flux background and we demonstrate that there is a large
flux double scaling limit where the action is dominated by a Chern-Simons-Myers
coupling term. A classical solution to this is the quantised spacetime known as
the quantum Nambu geometry. Various matrix models are obtained from this
action, these are the large flux dominated terms of the full actions for the
corresponding matrix models. The QNG gives rise to an expansion of D1-strings
to D4-branes in the IIA theory, so we obtain an action for the large flux terms
for this action which is verified by a dimensional reduction of the PST action
describing M5-branes. We make a generalisation of the D4-brane action to
describe M5-branes using a duality. We are describing the 3-form self-dual
field strength of a non-abelian generalisation of the PST action.Comment: PhD Thesis, 143 page