The concept of a smooth metric measure space has recently arisen as a useful object within Riemannian geometry, for example in Perelman's formulation of Ricci flow as a gradient flow. Taking two different perspectives, Lott, Villani, Sturm and Chang, Gursky and Yang have found two different approaches to defining a suitable notion of the Ricci curvature on a smooth metric measure space. While the formulations are different, they both include an extra dimensional parameter m which, in the limit m → ∞, both recover the curvatures that arise in Perelman's treatment of the Ricci flow. Associated to these curvatures are quasi-Einstein metrics, which are the appropriate notion of an Einstein metric in this setting. These include conformally Einstein metrics, the bases of Einstein warped products, and gradient Ricci solitons.

In this dissertation, we shall develop a treatment of smooth metric measure spaces which unites the perspectives of Lott-Villani-Sturm and Chang-Gursky-Yang. This treatment will include a definition of the Ricci and scalar curvature on these spaces, and the associated quasi-Einstein metrics. We shall then discuss three results which suggest that this notion is indeed the "correct" definition. First, we will see that quasi-Einstein metrics arise as critical points of the natural total scalar curvature functional, and moreover, that the restricted Yamabe-like problem allows one to understand the local geometry of a CWM, generalizing Perelman's entropies. Second, we will see that the notion of a CWM and their associated objects are well-behaved in the limit m → ∞, which will culminate with a nice Liouville-type theorem. Third, we will see that the space of noncollapsing compact quasi-Einstein metrics is precompact, so long as one allows the possibility of finitely many orbifold singularities. This final result can be viewed as an analog of the stability theorem for Ricci curvature lower bounds of Lott-Villani-Sturm to the setting of quasi-Einstein metrics.