Shrinking gradient Ricci solitons are models for the local geometry about a developing singularity under the Ricci flow. At present, all known examples of complete noncompact shrinkers are either locally reducible as products or possess conical structures at infinity. I will survey some recent results related to the problem of their classification including some joint work with Lu Wang in which we study the uniqueness of such asymptotic structures as a problem of parabolic unique continuation.