  We prove that every birationally superrigid Fano variety whose alpha
invariant is greater than (resp. no smaller than) $\frac{1}{2}$ is K-stable
(resp. K-semistable). We also prove that the alpha invariant of a birationally
superrigid Fano variety of dimension $n$ is at least $\frac{1}{n+1}$ (under
mild assumptions) and that the moduli space (if exists) of birationally
superrigid Fano varieties is separated.
