This paper reports an overview of recent DFT studies on the atomic scale characterization of Co(Ni)MoS active phases used in selective hydrodesulfurization (HDS) catalysis. A peculiar attention is paid to the effect of the sulfo-reductive reaction conditions on the stability and nature of the active sites present at the edges of the Co(Ni)MoS nano-crystallites. Combining DFT calculations and a thermodynamic model, it is shown that the promoter content at the crystallite edges in conjunction with the crystallite morphology may change with the sulfo-reductive conditions. We then show how the nature of the Co(Ni)MoS active sites impact the selective HDS of light sulfur compounds (alkylthiophene) versus olefin hydrogenation (alkylbutene). Combining DFT calculations of the adsorption of two relevant model molecules (2-methylthiophene and 2,3-dimethyl- butene) with the kinetic modelling of HDS/HydO selectivity, we propose an explanation on the origin of the selectivity based on the competitive adsorption of reactants.