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) ¹/₆s - W₊ ³ 0 {\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, \mathbbCP² \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.