The proposal is for unsaturated fatty substances with improved cold behavior obtained by (a) transesterifying a new low stearic (LS) sunflower oil with an oleic acid content of over 85 wt. % and a stearic acid content of under 3 wt. %, and (b) hydrogenating the resultant methyl esters in the prior art manner into the corresponding unsaturated fatty alcohols in the iodine number range from 90 to 100. The fatty substances are also distinguished by improved color and odor and are suitable for making fatty chemical products, e.g. alkoxylates, ether sulphates, sulphates and esters.