Research on the selective reduction of NOx with hydrocarbons under lean-burn conditions using non-zeolitic oxides and platinum group metal (PGM) catalysts has been critically reviewed. Alumina and silver-promoted alumina catalysts have been described in detail with particular emphasis on an analysis of the various reaction mechanisms that have been put forward in the literature. The influence of the nature of the reducing agent, and the preparation and structure of the catalysts have also been discussed and rationalised for several other oxide systems. It is concluded for non-zeolitic oxides that species that are strongly adsorbed on the surface, such as nitrates/nitrites and acetates, could be key intermediates in the formation of various reduced and oxidised species of nitrogen, the further reaction of which leads eventually to the formation of molecular nitrogen. For the platinum group metal catalysts, the different mechanisms that have been proposed in the literature have been critically assessed. It is concluded that although there is indirect, mainly spectroscopic, evidence for various reaction intermediates on the catalyst surface, it is difficult to confirm that any of these are involved in a critical mechanistic step because of a lack of a direct quantitative correlation between infrared and kinetic measurements. A simple mechanism which involves the dissociation of NO on a reduced metal surface to give N(ads) and O(ads), with subsequent desorption of N-2 and N2O and removal of O(ads) by the reductant can explain many of the results with the platinum group metal catalysts, although an additional contribution from organo-nitro-type species may contribute to the overall NOx reduction activity with these catalysts. It is concluded, after the investigation of hundreds of catalyst formulations, that many of the fundamental questions relating to lean deNO(x) reactions have been addressed and the main boundary conditions have been established. It seems clear that catalysts with sufficient activity, selectivity or stability to satisfy the demanding conditions that appertain in automotive applications are still far away. The rapidly growing interest in NOx storage systems reflects this situation, and it now seems to be the case that acceptable direct NOx reduction catalysts may be very difficult to find for lean-burn applications.