A complex projective variety is Q-Fano if it has klt singularities and the anti-canonical divisor is Q-Cartier and ample. Starting from dimension 2, the anti-canonical volume of a Q-Fano variety can be arbitrarily large, e.g. weighted projective spaces. Recently, Fujita showed that an n-dimensional Kahler-Einstein Q-Fano variety has volume at most (n+1)^n. In this talk, I will discuss a refinement of Fujita's volume upper bounds involving invariants of the local singularities. If time permits, I will also talk about an equivalent relation between K-semistability and de Fernex-Ein-Mustata type inequalities. Part of this work is joint with Chi Li. Speaker(s): Yuchen Liu (Princeton University)