NOVELTY - Production of biodiesel involves: esterification of the fatty acids of fatty-substances having iodine numbers greater than 135 gI2/100g, with specific alkyl alcohol, optionally in the presence of a suitable catalyst and selective catalytic hydrogenation of the obtained fatty acid alkyl esters, with a degree of unsaturation of less than or equal to 120 gI2/100 g and which are liquid at temperatures greater than or equal to -20 deg C.    USE - For the production of biodiesel (claimed).    ADVANTAGE - The method uses catalysts containing copper on suitable solid support for the hydrogenation of high iodine number fatty substances, good selectivity and activity are obtained. The method allows the use of high iodine number fatty substances as starting materials for the production of biodiesel, with the consequent advantages of using raw materials that are widely available and cheap, hence, providing a low cost fuel, the production cost of which is not influenced by problems associated with the marketing and/or disposal of glycerine, when the raw material used is tall oil; allows the recycling of the reagents and by-products, such as the alkyl alcohol, the catalysts and the water, all to the benefit of the environment. The catalyst used allows the removal of polyenes with the formation of monoenes, without in any way achieving the complete saturation of the molecule, with the formation of saturated fatty acid esters.    DETAILED DESCRIPTION - Production of biodiesel involves:    (1) esterification of the fatty acids of fatty-substances having iodine numbers greater than 135 gI2/100g, with a 1-10C alkyl alcohol, optionally in the presence of a suitable catalyst and selective catalytic hydrogenation of the obtained fatty acid alkyl esters, with a degree of unsaturation of less than or equal to 120 gI2/100 g and which are liquid at temperatures greater than or equal to -20 deg C.    INDEPENDENT CLAIMS are included for the following: use of a copper catalyst supported on Al2 O3, SiO2, sepiolite or TiO2 (preferably Al2O3, sepiolite or TiO2, especially Cu/Al2O3) for the selective hydrogenation of Tall oil fatty acid alkyl esters; and use of fatty substances having a iodine number greater than 135 (preferably greater than 140, especially greater than 150) gI2/100g, for the production of biodiesel.