In this talk, I will describe recent joint work with Brett Kotschwar in which we investigate the rigidity of asymptotically conical structures of shrinking Ricci solitons and show that, if two shrinking solitons are asymptotic to the same cone along some end of each, then the solitons must actually be isometric on some neighborhoods of infinity of these ends. As an application, we prove that the only complete shrinker asymptotic to a rotationally symmetric cone is the Gaussian soliton.