The present invention relates to a process for the production of polyunsaturated fatty acids in an organism by introducing, into the organism, nucleic acids which encode polypeptides with Δ;5-elongase activity. Advantageously, these nucleic acids can be expressed in the organism together with further nucleic acids which encode polypeptides of the biosynthesis of the fatty acid or lipid metabolism. Especially advantageous are nucleic acids which encode Δ;6-desaturases, Δ;5-desaturases, Δ;4-desaturases and/or Δ;6-elongases. These desaturases and elongases are advantageously derived from Thalassiosira, Euglena or Ostreococcus. The invention furthermore relates to a process for the production of oils and/or triacylglycerides with an elevated content of long-chain polyunsaturated fatty acids, and oils and/or triacylglycerides thus obtained. The invention also relates to the nucleic acids, and constructs, vectors and transgenic organisms comprising the same, as well as oils, lipids and/or fatty acids produced by the process according to the invention and to their use.