The classification of Riemannian manifolds with positive and non-negative sectional curvature is a long-standing problem in Riemannian

geometry. In this talk I will give an overview of known results and summarize joint work with Zheting Dong and Catherine Searle on the classification of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, almost isotropy-maximal torus action. This classification has many applications, in particular the Maximal Symmetry Rank conjecture holds for this class of manifolds.