The present invention relates to an improved process for the preparation of unsaturated fatty acids and to a process for the preparation of triglycerides with an increased content of unsaturated fatty acids. The invention relates to the generation of transgenic microorganism, with an increased content of fatty acids, oils or lipids with Δ6 double bonds owing to the expression of a moss Δ6-desaturase. The invention furthermore relates to transgenic organisms comprising a Δ6-desaturase gene, and to the use of unsaturated fatty acids ort of the triglycerides with an increased content if unsaturated fatty acids prepared in the process.