We explore the interactions between optimal mass transport theory and the geometry of the underlying (and other related) spaces. In particular, we present a sketch of the proof of Luis Caffarelli's contraction theorem and consider its geometric consequences and possible extensions, especially in the context of spherical geometry. We study also the concept of Hessian metrics --- one of the geometric tools implemented by Eugenio Calabi in his investigation of the properties of solutions of the general Monge-Ampere equation for the Euclidean space ( R n, dRn) --- and we summarize one of the significant contributions arising from Calabi's work as a lower Ricci curvature bound. In the end, we give an exposition of the author's recent results concerning modified Hessian pseudo-metrics. These results generalize a portion of Calabi's theory of Hessian metrics to n-dimensional space forms of constant positive sectional curvature and lead to new lower Ricci curvature bounds.

We emphasize throughout this dissertation those connections relating the theory of optimal mass transport and curvature bounds.