An NO.sub.x sorption-and-reduction type catalyst is made which includes a hydrogen generating catalyst in which Rh is loaded on a support including an Al.sub.2O.sub.3--ZrO.sub.2 composite oxide. Since the Al.sub.2O.sub.3--ZrO.sub.2 composite oxide exhibits a basicity lower than ZrO.sub.2, SO.sub.x are less likely to approach so that it is possible to suppress the sulfur poisoning of Rh. Moreover, the Al.sub.2O.sub.3--ZrO.sub.2 composite oxide exhibits higher heat resistance than ZrO.sub.2, and it has a function of improving the steam reforming reaction activity of Rh in the same manner as ZrO.sub.2. Therefore, the post-durability NO.sub.x purifying ability of the NO.sub.x sorption-and-reduction type catalyst is improved.