The present invention relates to an improved method for the specific production of unsaturated .omega.-3 fatty acids and a method for the production of triglycerides having an increased content of unsaturated fatty acids, in particular .omega.-3 fatty acids having more than three double bonds. The invention relates to the production of a transgenic organism, preferably a transgenic plant or a transgenic microorganism, having an increased content of fatty acids, oils or lipids having .DELTA.6 double bonds due to the expression of a .DELTA.6-desaturase from Primulaceae. The invention additionally relates to expression cassettes containing a nucleic acid sequence, a vector and organisms containing at least one nucleic acid sequence or an expression cassette. The invention further relates to unsaturated fatty acids and triglycerides having an increased content of unsaturated fatty acids and use thereof.