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 .DELTA.5-elongase, .DELTA.6-desaturase, .DELTA.5-desaturase, .DELTA.4-desaturase, .DELTA.12-desaturase and/or .DELTA.6-elongase activity. These desaturases and elongases are advantageously derived from 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. The invention furthermore relates to the nucleic acid sequences, nucleic acid constructs, vectors and organisms comprising the nucleic acid sequences according to the invention, to vectors comprising the nucleic acid sequences and/or the nucleic acid constructs and to transgenic organisms comprising the abovementioned nucleic acid sequences, nucleic acid constructs and/or vectors. A further part of the invention relates to oils, lipids and/or fatty acids produced by the process according to the invention and to their use. Moreover, the invention relates to unsaturated fatty acids and to triglycerides with an elevated content of unsaturated fatty acids and to their use.