  In this paper, we develop the crystal basis theory for the quantum queer
superalgebra $U_q(\mathfrak q(n))$. We define the notion of crystal bases and
prove the tensor product rule for $U_q(\mathfrak q(n))$-modules in the category
$O_int^{\geq 0}$. Our main theorem shows that every $U_q(\mathfrak
q(n))$-module in the category $O_int^{\geq 0}$ has a unique crystal basis.
