  This thesis consists of two parts. In the first part we investigate the
worldvolume supersymmetry algebra of multiple membrane theories. We begin with
a description of M-theory branes and their intersections from the perspective
of spacetime and worldvolume supersymmetry algebras. We then provide an
overview of the recent work on multiple M2-branes focusing on the
Bagger-Lambert theory and its relation to the Nambu-Poisson M5-brane and the
ABJM theory. The worldvolume supersymmetry algebras of these theories are
explicitly calculated and the charges interpreted in terms of spacetime
intersections of M-branes. The second part of the thesis looks at $l_p^3$
corrections to the supersymmetry transformations of the Bagger-Lambert theory.
We begin with a review of the dNS duality transformation which allows a gauge
field to be dualised to a scalar field in 2+1 dimensions. Applying this duality
to the $\alpha'^2$ terms of the D2-brane supersymmetry transformations we are
able to determine the $l_p^3$ corrections to the supersymmetry transformations
of the scalar and fermion fields for the abelian theory. Generalising to the
`non-abelian' Bagger-Lambert theory we are able to determine the $l_p^3$
correction to the supersymmetry transformation of the fermion field. Along the
way make a number of observations relating to the implementation of the dNS
duality transformation at the level of supersymmetry transformations.
