A remarkable feature of the Ricci flow in dimension three is the Hamilton-Ivey curvature pinching estimate discovered in early 90s. This estimate played a key role in the study of 3D Ricci flow singularity models. In this talk, l will discuss generalizations of the Hamilton–Ivey curvature pinching estimate for Ricci solitons, which are self-similar solutions to the Ricci flow, in dimension three and higher. In particular, I'll present new Hamilton–Ivey-type curvature pinching estimates we found recently for asymptotically conical gradient expanding Ricci solitons. It is based on joint work with Junming Xie.