  We construct and investigate the structure of the Khovanov-Lauda-Rouquier
algebras $R$ and their cyclotomic quotients $R^\lambda$ which give a
categrification of quantum generalized Kac-Moody algebras. Let $U_\A(\g)$ be
the integral form of the quantum generalized Kac-Moody algebra associated with
a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the
Grothedieck group of finitely generated projective graded $R$-modules. We prove
that there exists an injective algebra homomorphism $\Phi: U_\A^-(\g) \to
K_0(R)$ and that $\Phi$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$.
Let $B(\infty)$ and $B(\lambda)$ be the crystals of $U_q^-(\g)$ and
$V(\lambda)$, respectively, where $V(\lambda)$ is the irreducible highest
weight $U_q(\g)$-module. We denote by $\mathfrak{B}(\infty)$ and
$\mathfrak{B}(\lambda)$ the isomorphism classes of irreducible graded modules
over $R$ and $R^\lambda$, respectively. If $a_{ii}\ne 0$ for all $i\in I$, we
define the $U_q(\g)$-crystal structures on $\mathfrak{B}(\infty)$ and
$\mathfrak{B}(\lambda)$, and show that there exist crystal isomorphisms
$\mathfrak{B}(\infty) \simeq B(\infty)$ and $\mathfrak{B}(\lambda) \simeq
B(\lambda)$. One of the key ingredients of our approach is the perfect basis
theory for generalized Kac-Moody algebras.
