A fascinating phenomena on a closed n-manifold of positive sectional curvature is that the volume is not small when n is even while the volume can be arbitrarily small i.e., collapsed when n is odd (after normalizing the curvature bounded above by one). In this talk, we will discuss a classification of collapsed 5-manifolds of pinched positive sectional curvature, and its link to the classification of positively curved 5-manifolds (non-simply connected) which admit compactible local isometric torus actions. This is a joint work with Fuquan Fang.