  In this paper, we prove Khovanov-Lauda's cyclotomic categorification
conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the
quantum group associated with a symmetrizable Cartan datum and let $V(\Lambda)$
be the irreducible highest weight $U_q(g)$-module with a dominant integral
highest weight $\Lambda$. We prove that the cyclotomic Khovanov-Lauda-Rouquier
algebra $R^{\Lambda}$ gives a categorification of $V(\Lambda)$.
