Khovanskii-Teissier inequality in complex geometry is a profound analog of Alexandrov-Fenchel inequality in convex geometry. In this talk, we shall present some new Khovanskii-Teissier type inequalities in mixed and degenerate settings, basing on developing new Hodge-Riemann bilinear relation on compact Kahler manifolds. Time permitting, characterization of the equality in the boundary case will also be mentioned.