Normalized volume of a klt singularity and K-stability

Normalized volume is a certain functional on the space of valuations centered at x, where (X, x) is a klt singularity. Recently numerous works studying the minimization problem for this functional have appeared. This problem turns out to be important both for local theory of klt singularities and for some global properties of Fano varieties, like K-semistability. In my talk I will give an introduction to this subject and discuss an application of this theory to K-stability of cubic threefolds.