Hydrodesulfurization (HDS) and hydrodenitrogenation (HDN) of petroleum products and intermediates are reviewed to provide the basis for developing processes to produce gasoline and diesel oil with very low sulfur content. The reactivity, selectivity and inhibition (susceptibility of substrate molecules to inhibitors) in the catalytic process are very important to develop efficient processes. Recent advances in the understanding of active species, supports and supporting methods are also critically reviewed to suggest the design of catalysts with adequate activity to satisfy future regulations on transportation fuels. Details of the structures of the catalysts are not discussed, but the mechanisms of hydrodesulfurization and inhibition are summarized. Catalyst deactivation and reactor design are also briefly reviewed. New approaches to achieve deep hydrodesulfurization are proposed.