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