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It has previously been proposed that the the theory of strings and branespossesses 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 fourdimensions possesses a large symmetry group generated by the Kac-Moody algebra$A_1^{+++}$. These symmetry groups predict the existence of an infinitecollection of fields in each theory, including a field that can be interpretedas a dual graviton. In this thesis we will discuss developments related tothese approaches to string theory and gravity.  We begin by reviewing the method of nonlinear realizations, then introducingthe Kac-Moody algebra $A_1^{+++}$, along with its first fundamentalrepresentation representation which introduces space-time, to low levels. Wewill then consider a nonlinear realization involving their semi-direct product,and use the associated Cartan forms to construct dynamics for the low levelfields of this nonlinear realization. We will then propose duality relationsbetween fields of the theory at different levels, and propose a nonlinearequation of motion for the dual graviton arising from $A_1^{+++}$.  We then consider the analogous results for $E_{11}$, and describe a proposednonlinear dual graviton equation in the context of string theory. We nextdiscuss irreducible representations of $E_{11}$. We will construct isotropygroups associated to massive and massless particles, branes such as the M2 andM5 branes, and the IIA string. Finally we present an approach for introducingspinorial representations of the Kac-Moody algebra $E_{11}$.

Applications of Kac-Moody Algebras to Gravity and String Theory