  We provide two methods of producing the $Q$-operator of XXZ spin chain of
higher spin, one for $N$th root-of-unity $q$ with odd $N$ and another for a
general $q$, as the generalization of those known in the six-vertex model. In
the root-of-unity case, we discuss the functional relations involving the
constructed $Q$-operator for the symmetry study of the theory. The $Q$-operator
of XXZ chain of higher spin for a generic $q$ is constructed by extending
Baxter's argument in spin-1/2 case for the six-vertex $Q$-operator.
