Abstract: An Einstein metric, which arises naturally in general relativity and differential geometry, is a canonical structure with its Ricci curvature proportional to the metric. It is known that there is some obstruction to the existence of an Einstein metric in dimension four. More specifically, a long-standing conjecture states that, aside from a few symmetric examples, there are no other Einstein four-dimensional manifolds (E4M) with non-negative sectional curvature. In this talk, we discuss recent progress on this conjecture. The main results classify E4Ms with some pinched sectional curvature conditions and the proof involves a novel application of the Ricci flow theory. This is a joint work with Xiaodong Cao.