Abstract: The second kind of curvature operator has received recent attention following the resolution of Nishikawa's conjecture by Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and myself. These works have shown that a closed Riemannian manifold with 3-nonnegative curvature operator of the second kind is either flat or diffeomorphic to a spherical space form. In this talk, I will briefly introduce the curvature operator of the second kind and talk about the proof of Nishikawa's conjecture. Then I will talk about my recent work on new sphere theorems under negative lower bounds of the curvature operator of the second kind.