In this thesis, we define the 𝛿-invariant for log Fano cone singularities, and show that the necessary and sufficient condition for K-semistability is 𝛿 ≥ 1. This generalizes the result of C. Li and K. Fujita. We also prove that on any log Fano cone singularity of dimension 𝑛 whose 𝛿-invariant is less than (𝑛+1)/𝑛, any valuation computing 𝛿 has a finitely generated associated graded ring. This shows a log Fano cone is K-polystable if and only if it is uniformly K-stable. Together with earlier works, this implies the Yau-Tian-Donaldson Conjecture for Fano cone.Ph.D.