An exhaust gas purifying catalyst contains copper oxide, ZSM-5 and/or zeolite .beta., and an oxide of magnesium and/or calcium is further added thereto. A process for purifying an exhaust gas using such a catalyst is also included. By using the catalyst, NOx in an exhaust gas containing much oxygen such as exhaust gas of diesel engines can be efficiently removed and oxidation of SO.sub.2 under high-temperature conditions can be suppressed.