A selective overview of recent studies on both three-way and deNO(x) catalysis (we shall note 'deNO(x)', the removal of NO in the presence of an excess of oxygen) leads to an unique and general model of these reactions, based on kinetic concepts. Two kinds of active sites have to be first defined: cationic and zero-valent metal ones. The first type can form either through the reduction of the support ((v))-Ce3+-((v)) or (ii) a strong metal-support interaction ((v))-Rh+-((v))/CeO2, both linked to adjacent oxygen vacancies ((v)) of the reducible support, or (iii) a surface transition metal (TM) complex (TM in zeolite for instance). The second kind of sites are accessible supported-zero-valent noble metal atoms. These two kinds of sites are involved in three-way Catalysts (TWC), whereas only the cationic ones concern deNO(x) reactions. Therefore, nitric oxide chemisorption can be either 'associative' on the first kind of sites - leading to dinitrosyl or hyponitrite species - or 'dissociative', on the second ones, leading to oxygen and nitrogen atoms adsorbed on the sites. Two different catalytic sequences of elementary steps can then be defined. On cationic sites, successive N-O bond scissions of dinitrosyl or hyponitrite species occur, potentially able to produce intermediate N2O, and in all cases leaving oxygen atoms adsorbed on the active site, and inhibiting a further adsorption of NO. A reductant is then necessary to remove these oxygen atoms and permit the reaction to proceed further. On zero-valent metallic sites, at the temperature of reaction, NO suffers a dissociative chemisorption leading again to surface oxygen atoms. Again, a reductant is necessary to remove these oxygen species and permit the reaction to proceed again. In this first paper, TWC are considered and the reductant is CO. Two catalytic cycles are considered based on our results on temperature-programmed desorption and surface reactions of NO in a stoichiometric CO/NO/O-2 mixture.