The present invention relates to the production of .gamma.-linolenic acid (18:3.sup..DELTA.6,9,12) or stearidonic acid (18:4.sup..DELTA.6,9,12,15) or .gamma.-linolenic acid (18:3.sup..DELTA.6,9,12) and stearidonic acid (18:4.sup..DELTA.6,9,12,15) in transgenic plants of the family Brassicaceae, where the transgenic plants comprise at least 10% by weight of oleic acid based on the total fatty acid content and, as the result of the activity of the .DELTA.6-desaturases used in the method, have an increased .DELTA.6-C.sub.18-fatty acid content. The invention furthermore relates to novel nucleic acid sequences which code for the .DELTA.6-desaturases used in the method, gene constructs comprising these nucleic acid sequences, a vector and transgenic plants comprising at least one nucleic acid sequence or a gene construct.