  The purpose of this thesis is to explore the properties of multiple
coincident M2- and M5-branes. We begin with a review of the BLG and ABJM models
of multiple M2-branes and our focus will be on their formulation in terms of
3-algebras. We then examine the coupling of multiple M2-branes to the
background 3-form and 6-form gauge fields of eleven-dimensional supergravity.
In particular we show in detail how a natural generalisation of the Myers
flux-terms, along with the resulting curvature of the background metric, leads
to mass terms in the effective field theory. Working to lowest nontrivial order
in fermions, we demonstrate the supersymmetric invariance of the
four-derivative order corrected Lagrangian of the Euclidean BLG theory and
determine the theory's higher derivative corrected supersymmetry
transformations. The supersymmetry algebra is also shown to close on the scalar
and gauge fields. We also consider periodic arrays of M2-branes in the ABJM
model in the spirit of a circle compactification to D2-branes in type IIA
string theory. The result is a curious formulation of three-dimensional
maximally supersymmetric Yang-Mills theory. Upon further T-duality on a
transverse torus we obtain a non-manifest-Lorentz-invariant description of
five-dimensional maximally supersymmetric Yang-Mills which can be viewed as an
M-theory description of M5-branes on $\mathbb{T}^3$. After reviewing work to
describe multiple M5-branes using 3-algebras we show how the resulting novel
system of equations reduces to one-dimensional motion on instanton moduli
space. Quantisation leads to the previous light-cone proposal of the (2,0)
theory, generalised to include a potential that arises on the Coulomb branch as
well as couplings to background gauge and self-dual 2-form fields.
