Abstract: K-stability is the algebraic notion which is supposed to characterize whether a Fano variety admits a Kähler-Einstein metric. One important feature of the notion K-stability is that it is supposed to give a nicely behaved moduli space. To construct the K-moduli space of Q-Fano varieties as an algebraic space, one important step is to prove the openness of K-(semi)stable locus in families. In this talk, I will explain the proof of openness of uniform K-stability in families of Q-Fano varieties. This is achieved via showing the lower semi-continuity of delta-invariant, an interesting invariant introduced by Fujita and Odaka similar to Tian's alpha-invariant. This is joint work with Harold Blum.