This thesis first gives reviews of the theories of spinors and manifolds, and discusses a formalism in which spinorial, vectorial and tensorial fields may be represented upon a manifold. In particular, the Riemann-Cartan manifold is defined; the sphere as a Riemann-Cartan manifold is given as an example, and the effects of the geometry of such a manifold are discussed. The field equations of Einstein-Cartan theory, which treats spacetime as a Riemann-Cartan manifold, are then derived; the macroscopic limit is considered and shown to reproduce the Einstein equation obtained from general relativity. An introduction to the background field method approach to quantum field theories is then given; in particular, the metric and vierbein background field method approaches to the quantum formulation of Einstein-Cartan theory are discussed. The Faddeev-Popov method of gauge-fixing is then discussed, and the propagators of the graviton and Feynman-De Witt-Faddeev-Popov ghosts are derived in a general gauge. The coupling of the standard model matter fields to gravity is then discussed; in particular, the coupling of scalars and fermions to gravity is considered, and the tree-level Feynman rules are derived. Simple scattering processes are considered and shown to be gauge-independent, and to reproduce the Newtonian potential in the non-relativistic limit.