Abstract: Riemannian manifolds with special geometric properties which are also invariant under a torus action have been studied frequently. We will first review some known results in any dimension if the metric has non-negative sectional curvature. We are interested not only in the diffeomorphism type of the manifold, but also to identify the group action. Previous results of this type are only known in dimension 4. We will discuss some recent joint work with Karsten Grove on 5 dimensional Riemannian manifolds with non-negative curvature invariant under a torus action. Some of these are exotic actions.