First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,j R2ij [les ] r2/(n−1), where [sum ]i,j R2ij is the squared norm of the Ricci curvature tensor.