The deep hydrodesulfurization (HDS) of diesel fuels requires the decomposition of refractory compounds such as 4,6-dimethyldibenzothiophene (46DMDBT). There is a general agreement on the fact that the low reactivity of these compounds is due to steric hindrance of the transition state leading to C-S bond cleavage, which annihilates the effect of the promoter to a large extent. The consequence is that their so-called "direct desulfurization pathway" is particularly inhibited.Various issues were considered to circumvent the difficulty to eliminate the HDS-resistant molecules and therefore to reach deep desulfurization. Two of them at least consist in designing new hydrotreating catalysts (in addition to the improvement of the alumina-supported conventional catalysts): (i) catalysts with improved hydrogenation properties; (ii) bifunctional catalysts containing an acid component. The main findings obtained with the first class of catalysts are summarized. On these catalysts HDS was found very sensitive to the inhibition by aromatics. The studies regarding the second category of catalysts are analyzed in more detail. Several techniques were used to introduce acid components, including mesoporous materials, in hydrotreating catalysts: physical mixing, binding, deposition of sulfide precursors on an acid-alumina support, deposition of sulfide precursors on an acidic support. Most of these catalysts were found more active than conventional catalysts in the HDS of compounds such as 46DMDBT. The various interpretations of the effect of the acid component are reported and discussed. It is however generally accepted that the improvement of the reactivity, on this category of catalysts, of the HDS-resistant compounds is due to their acid-catalyzed isomerization and disproportionation into more reactive derivatives. When acidic materials were used directly as supports it was difficult to obtain a good association of, for instance, molybdenum, with promoters. Nevertheless, in certain cases catalysts were obtained which were more active than conventional catalysts in the HDS of compounds such as 46DMDBT or of gas oils containing such impurities. However although the catalysts containing acid components proved efficient in hydrotreating various kinds of oils, they suffer several drawbacks such as deactivation by coke deposition and inhibition of HDS by aromatics. Moreover nitrogen impurities which inhibit HDS even with conventional catalysts may also impede seriously their use in practice. (C) 2003 Elsevier B.V. All rights reserved.