The Riemann curvature tensor on a Riemannian manifold induces two
kinds of curvature operators: the first kind acting on two-forms and the
second kind acting on (traceless) symmetric two-tensors. The curvature
operator of the second kind recently attracted a lot of attention due to the
resolution of Nishikawa's conjecture by X.Cao-Gursky-Tran and myself. In
this talk, I will survey some recent works on the curvature operator of the
second kind on Riemannian and Kahler manifolds and also mention some
interesting open problems. The newest result, joint with Harry Fluck at
Cornell University, is an investigation of the curvature operator of the
second kind in dimension three and its Ricci flow invariance.