The invention relates to an improved method for the specific production ofunsaturated .omega.-3 fatty acids and a method for producing triglyceridescontaining an increased number of unsaturated fatty acids, especially .omega.-3 fatty acids comprising more than three double bonds. Disclosed is theproduction of a transgenic organism, preferably a transgenic plant, or atransgenic microorganism containing an increased number of fatty acids, oils,or lipids comprising .DELTA.-4 double bonds as a result of a .DELTA.-4desaturase being expressed from Euglena gracilis. The invention furtherrelates to expression cassettes containing a nucleic acid sequence, a vector,and organisms containing at least one nucleic acid sequence or an expressioncassette. Also disclosed are unsaturated fatty acids and triglycerides havingan increased fatty acid content and the use thereof. Fatty acids andtriglycerides are used in a plurality of application in the food industry, inanimal nutrition, cosmetics, and pharmaceuticals and are suitable for manydifferent applications depending on whether free saturated or unsaturatedfatty acids or triglycerides having an increased saturated or unsaturatedfatty acid content are used.