  It has previously been proposed that the the theory of strings and branes
possesses a large symmetry group generated by the Kac-Moody algebra $E_{11}$.
It has also previously been proposed that the the theory of gravitation in four
dimensions possesses a large symmetry group generated by the Kac-Moody algebra
$A_1^{+++}$. These symmetry groups predict the existence of an infinite
collection of fields in each theory, including a field that can be interpreted
as a dual graviton. In this thesis we will discuss developments related to
these approaches to string theory and gravity.
  We begin by reviewing the method of nonlinear realizations, then introducing
the Kac-Moody algebra $A_1^{+++}$, along with its first fundamental
representation representation which introduces space-time, to low levels. We
will then consider a nonlinear realization involving their semi-direct product,
and use the associated Cartan forms to construct dynamics for the low level
fields of this nonlinear realization. We will then propose duality relations
between fields of the theory at different levels, and propose a nonlinear
equation of motion for the dual graviton arising from $A_1^{+++}$.
  We then consider the analogous results for $E_{11}$, and describe a proposed
nonlinear dual graviton equation in the context of string theory. We next
discuss irreducible representations of $E_{11}$. We will construct isotropy
groups associated to massive and massless particles, branes such as the M2 and
M5 branes, and the IIA string. Finally we present an approach for introducing
spinorial representations of the Kac-Moody algebra $E_{11}$.
