Arguably the two most successful physical theories of the last century are Einstein's theory of general relativity, which tells us how objects behave at very large scales, and quantum field theory, which describes the small scale behavior of matter. Although both general relativity and quantum field theory provide highly accurate descriptions of the Universe in the realms in which they apply, attempts to unify the two theories have so far been unsuccessful. While various candidate theories of quantum gravity exist, due to the mathematical complexity involved and the relative inaccessability of the quantum gravity regime to direct experiment we do not yet know which, if any, of these candidates is correct. Stochastic gravity is an attempt to bridge (or at least narrow) the gap between these theories by using statistical techniques to incorporate the quantum behavior of matter into Einstein's field equations for gravity. The central object of interest within the framework of stochastic gravity is a quantity called the noise kernel which describes how the quantum fluctuations of matter are correlated across spacetime; studying the noise kernel allows us to investigate how the quantum fluctuations affect the geometry of spacetime. My work during the course of this program involved the calculation of the noise kernel in an exponentially expanding spacetime known as de Sitter space. This spacetime is of interest for two reasons: first, during the early history of the universe, it is believed that the universe underwent a period of rapid exponential expansion, and second, it appears that exponential expansion may describe the end of the universe as well. Thus, investigating quantum effects in such a spacetime may give us insight into the past and future evolution of our universe.