The present invention relates to the production of γ-linolenic acid (18:3Δ6,9,12) or stearidonic acid (18:4Δ6,9,12,15) or γ-linolenic acid (18:3Δ6,9,12) and stearidonic acid (18:4Δ6,9,12,15) 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 Δ6-desaturases used in the method, have an increased Δ6-C18-fatty acid content. The invention furthermore relates to novel nucleic acid sequences which code for the Δ6-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.