The present invention relates to a process for the production of arachidonic acid (=APA) or eicosapentaenoic acid (=EPA) or arachidonic acid and eicosapentaenoic acid, advantageously in the seed of transgenic plants of the family Brassicaceae with a content of ARA or EPA or ARA and EPA of at least 3% by weight based on the total lipid content of the transgenic plant, by introducing, into the organism, nucleic acids which code polypeptides with .DELTA.6-desaturase, .DELTA.6-elongase and .DELTA.5-desaturase activity, where, as the result of the enzymatic activity of the introduced enzymes, a fatty acid selected from the group consisting of the fatty acids oleic acid [C18:1.sup..DELTA.9], linoleic acid [C18:2.sup..DELTA.9, 12], .alpha.-linolenic acid [C18:3.sup..DELTA.6, 9, 12], icosenoic acid (20:1.sup..DELTA.11) and erucic acid [C22:1.sup..DELTA.11] is reduced by at least 10% in comparison with the nontransgenic wild-type plant. Advantageously, further enzymes selected from the group of the enzymes (.omega.3-desaturases, .DELTA.12-desaturases, .DELTA.6-desaturases, .DELTA.6-elongases, .DELTA.5-desaturases, .DELTA.5-elongases and/or .DELTA.4-desaturases can be introduced into the plants.