The mechanism of NOx decomposition is reviewed. This reaction is first studied on metal or oxide single crystals, in order to get solid basis from catalysis on well-defined surface structures. If we understand few things about this reaction, at least the catalytic surface is known. This statement is not 100% correct, as the surface is reconstructed during the catalytic reaction; and this point is considered in the manuscript. Then we analyse this reaction on metallic catalysts supported on metal oxides. The crystallites are more or less dispersed, and the support can play the role of a modifier or of a promoter. A general consensus emerges on (i) the chemisorption of the reactant(s) which causes adsorbate-induced restructuring of the surfaces, (ii) the substrate, which is influenced by the redox properties of the support and by the particle sizes of the metallic aggregates, (iii) the fact that low-coordinated surface atoms are the more active in N-O bond breaking, and (iv) the general pathways of the reaction, where redox mechanisms involving nitrosyl species, as intermediates, moving to nitrite-nitrosyl or dinitrosyl species are proposed. All these points are discussed.