The invention relates to a method for the production of multiply-unsaturated fatty acids in an organism, into which nucleic acids have been introduced, which code for polypeptides with .DELTA.-5 elongase activity. Said nucleic acid sequences, optionally with further nucleic acid sequences, coding for polypeptides for the biosynthesis of fatty acids and lipid metabolism, are advantageously expressed in the organism. Nucleic acid sequences coding for a .DELTA.-6 desaturase, a .DELTA.-5 desaturase, .DELTA.-4 desaturase and/or .DELTA.-6 elongase activity are particularly advantageous and, advantageously, said saturases and elongases are derived from Thalassiosira, Euglena or Ostreococcus. The invention further relates to a method for the production of oils and/or triacylglycerides with an increased content of long-chain, multiply-unsaturated fatty acids. A particular embodiment of the invention is a method for the production of unsaturated .omega.-3 fatty acids and a method for the production of triglycerides with an increased content of unsaturated fatty acids, in particular, of .DELTA.-3 fatty acids with more than three double bonds. Also disclosed is the production of a transgenic organism, preferably a transgenic plant, or a transgenic microorganism with increased content of .omega.-3 fatty acids, oils or lipids with .omega.-3 double bonds as a result of the expression of the elongases and desaturases employed in the above method, preferably in combination with .omega.-3 desaturases, for example a .omega.-3 desaturase from fungi of the family Pythiaceae such as the genus Phytophtora, for example, the genus and species Phytophtora infestans, or a .omega.-3 desaturase from algae such as the family Prasinophyceae, for example, the genus Ostreococcus and, particularly, the genus and species Ostreococcus tauri or diatomaceae such as the genus Thalassiosira and, particularly, the genus and species Thalassiosira pseudonana. The invention also relates to the nucleic acid sequences, nucleic acid constructs, vectors and organisms containing the nucleic acid sequences and/or the nucleic acid constructs and transgenic organisms containing said nucleic acid sequences, nucleic acid constructs and/or vectors. A further part of the invention relates to oils, lipids and/or fatty acids produced according to the above method and use thereof and, furthermore, unsaturated fatty acids and triglycerides with an increased content of unsaturated fatty acids and use thereof.