n this talk, we will discuss the uniqueness of K-polystable log Fano cone degeneration of any K-semistable log Fano cone. This confirms a conjecture of Donaldson-Sun stating that the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kaehler- Einstein Fano manifolds depends only on the algebraic structure of the singularity. (this is a joint work with Chi Li and Chenyang Xu)