  We algebraically prove K-stability of polarized Calabi-Yau varieties and
canonically polarized varieties with mild singularities. In particular, the}
"stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form
compact moduli space, are proven to be K-stable although it is well known that
they are \textit{not} necessarily asymptotically (semi)stable. As a
consequence, we have orbifold counterexamples, to the folklore conjecture
"K-stability implies asymptotic stability". They have Kahler-Einstein
(orbifold) metrics so the result of Donaldson does not hold for orbifolds.
