We derive new identities for the Weyl tensor of gradient Ricci solitons, specifically in dimension four. For instance, we obtain a Bochner- Weitzenbock-type formula for the norm of the self-dual Weyl tensor which has some applications including a connection between the geometry and topology. The proof uses some ideas related to Einstein manifolds and the solution of the Yamabe problem. We are also concerned with the interaction of different components of curvature with the potential function. The Weyl tensor arises naturally in these investigations. This is a joint work with Xiaodong Cao.