  In this paper, we develop the crystal basis theory for quantum generalized
Kac-Moody algebras. For a quantum generalized Kac-Moody algebra $U_q(\mathfrak
g)$, we first introduce the category $\mathcal O_{int}$ of $U_q(\mathfrak
g)$-modules and prove its semisimplicity. Next, we define the notion of crystal
bases for $U_q(\mathfrak g)$-modules in the category $\mathcal O_{int}$ and for
the subalgebra $U_q^-(\mathfrak g)$. We then prove the tensor product rule and
the existence theorem for crystal bases. Finally, we construct the global bases
for $U_q(\mathfrak g)$-modules in the category $\mathcal O_{int}$ and for the
subalgebra $U_q^-(\mathfrak g)$.
