Infinite derivative theory of gravity is a modification to the general theory
of relativity. Such modification maintains the massless graviton as the only
true physical degree of freedom and avoids ghosts. Moreover, this class of
modified gravity can address classical singularities.

In this thesis some essential aspects of an infinite derivative theory of
gravity are studied. Namely, we considered the Hamiltonian formalism, where
the true physical degrees of freedom for infinite derivative scalar models
and infinite derivative gravity are obtained. Furthermore, the Gibbons-Hawking-York
boundary term for the infinite derivative theory of gravity was obtained.
Finally, we considered the thermodynamical aspects of the infinite derivative
theory of gravity over different backgrounds. Throughout the thesis, our
methodology is applied to general relativity, Gauss-Bonnet and f(R) theories
of gravity as a check and validation.