A new approach to the active sites of Mo-containing catalysts for hydrodesulfurization and hydrodenitrogenation was studied by three methods, i.e., the inverse problem, fractal and site-type analyses. The distribution of the active sites of the sulfided and nitrided Mo/Al2O3 catalysts was studied based on the inverse problem using the Fredholm integral equation. The two active sites of the nitrided Mo/Al2O3 catalyst have a distribution involving the reaction rate and population; one site for a fast rate and large population and the other site for a slow rate and small population. The fractal analysis was applied to the morphology of the Mo nitride and the reactivity for the carbazole hydrodenitrogenation and related to the Mo-0 or Mo delta+ species. The fractal dimension of the Mo nitride surfaces was calculated to be from 2.02 to 2.60 and estimated the two active sites for the hydrogenation and denitrogenation. These sites were related to the Mo3+ of gamma-Mo2N from the XPS analysis. The active sites for the hydrodesulfurization of 4,6-dimethyldibenzothiophene on a sulfided NiMo/Al2O3 catalyst were discussed based on the site type analysis. A third active site of the sulfided NiMo/Al2O3 catalyst was proposed for the simultaneous desulfurization and hydrogenation from 4,6-dimethyldibenzothiophene to 3,3-dimethylbicyclohexyl as well as the two active sites for the hydrogenation to 3-methylcyclohexyltoluene and the desulfurization to 3,3-dimethylbiphenyl. (C) 2015 Elsevier B.V. All rights reserved.