This thesis details homeomorphism classi cation theorems for closed, simply connected, Riemannian 4-manifolds admitting an isometric circle action. We proceed by imposing progressively weaker curvature conditions, which results in a slight weakening of the classi cation. In particular, we consider positive, non-negative, and almost non-negative sectional curvature. For positive curvature, the homeomorphism classi cation is obtained by Hsiang and Kleiner in [24] and the equivariant di eomorphism classi cation by Grove and Searle in [18] and Grove and Wilking in [19]. For non-negative sectional curvature, we look to the independent work of Kleiner in his thesis [25], and Searle and Yang in [35] for the homeomorphism classi cation, which was then improved to a di eomorphism classi cation by work of Galaz-Garc a [13], and Grove and Wilking in [19]. For almost non-negative curvature, the recent work of Harvey and Searle in [22] gives the di eomorphism classi cation, which coincides with the classi cation for non-negatively curved manifolds. In fact, they show that almost non-negative curvature in this setting implies the existence of an S1-invariant metric of non-negative curvature.