The present invention relates to a process for producing arachidonic acid and/or eicosapentaenoic acid in transgenic useful plants by introducing into the plant nucleic acids coding for polypeptides having Δ6-desaturase, Δ6-elongase or Δ5-desaturase activity. Furthermore, a gene coding for an ω3-desaturase is advantageously expressed in said useful plants. In another advantageous embodiment of the process, further nucleic acid sequences coding for polypeptides of fatty acid or lipid metabolism biosynthesis may be expressed in the plants. Particularly advantageous nucleic acid sequences here are those coding for a Δ8-desaturase, Δ12-desaturase, Δ15-desaturase, Δ4-desaturase, Δ9-elongase and/or Δ5-elongase activity.The invention further relates to the use of the oils, lipids and/or fatty acids produced in the process according to the invention in feedstuffs or foodstuffs, cosmetics or pharmaceuticals.