It is well known that there exist several differentiable or topological obstructions to compact manifolds admitting metrics of positive scalar curvature. On the other hand, the family of manifolds with positive scalar curvature is quite large since any finite connected sum of them is still a manifold admitting a metric of positive scalar curvature. This talk is concerned with the classification question to this family.
         The classical uniformization theorem implies that a two-dimensional compact manifold with positive scalar curvature is diffeomorphic to the sphere or the real projective space. The combination of works of Scheon-Yau and Perelman gives a complete classification to compact three-dimensional manifolds with positive scalar curvature. In this talk we will discuss how to extend Schoen-Yau-Perelman's classification to four-dimension. This is based on the joint works with Bing-Long Chen and Siu-Hung Tang.