A key paradigm in geometry holds that control on the curvature of a space yields control on other geometric and topological invariants, and engenders new relationships between them. In this dissertation, we prove new results to this end. Our main invariant of focus is the min-max \emph{width} originally studied by Birkhoff, Almgren-Pitts, Simon-Smith, etc. Over the last century, a long roster of researchers has studied this invariant in its many forms, and how it interacts with curvature bounds. In a beautiful paper, Marques-Neves (2012) investigated how the min-max width of a Riemannian 3-sphere, realized as the area of an unstable minimal 2-sphere inside of it, is controlled in terms of a scalar curvature lower bound. They show that this width has a sharp upper bound which, when achieved, implies that the entire 3-sphere must be isometric to the standard round 3-sphere. Our main results here seek to \emph{stabilize} this theorem in various ways. Indeed, we ask what can be said of the 3-sphere when its width is \emph{almost-maximal}, providing an effective generalization of Marques-Neves' rigidity theorem. In Sweeney-Stufflebeam (2024), reported in Chapter 3.3, we thereby address a conjecture of Marques-Neves (cf. Sormani (2017)). In Chapter 3.4 we outline a framework for fully resolving another form of the conjecture. In Maximo-Stufflebeam (2025), reported in Chapter 3.2, the author and Davi M'aximo fully resolve the conjecture in 2-dimensions, thereby stabilizing a celebrated result of Toponogov. In the final Chapter 4 reporting our work in Stufflebeam (2023), we investigate a related problem informed by mathematical general relativity; we prove a stable version of the famous Min-Oo Conjecture in the only case where it is generally true---in dimension 2.