In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W + is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S 2 × S 2 with both (i) K > 0 and (ii) $$ {\textstyle{1 \over 6}}s - W_ + \ge 0 $$ . We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M 4 admits a metric g of non-negative curvature operator, then M 4 is one of S 4, $$ \mathbb{C}\rm P^2$$ and S 2 × S 2”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.