The catalyst structure of the present invention for purifying an exhaust gas is preferable for increasing the contact of an exhaust gas, to be treated, with a catalyst by disturbing the flow of the exhaust gas in a gas flow passage thereby obtain a highly efficient and compact apparatus for treating the exhaust gas. Such catalyst structure is produced by forming two or more catalyst elements each supporting a catalyst component on its surface and having flat plate portions and level-changing portions formed alternately therein with the angle formed between the flat plate portion and the level-changing portion being in a specific range, and then stacking the catalyst elements in a frame. A catalyst structure is also obtained by stacking a large number of the catalyst elements described above through metallic, ceramic, or glass netlike members interposed therebetween and each having a large number of perforated holes. The catalyst structure has such advantages that dust is hardly deposited between the catalyst elements, that a sufficiently high strength can be obtained even when the thickness of catalyst elements is reduced, and that the catalyst structure can be placed in an exhaust gas flue to use as an exhaust gas treating apparatus. When the exhaust gas treating apparatus is employed, an efficient exhaust gas treatment can be effected without damaging the catalyst elements while repressing the pressure loss.