The present invention relates to eukaryotic cells which have the ability to isomerize xylose directly into xylulose by transformation with nucleotide sequences encoding a xylose isomerase that has one or more specific sequence elements typical for isomerases which are functionally expressed in yeasts, such as xylose isomerases obtainable from bacteria of the genera Clostridium and Fusobacterium or a tunicate from the genus Ciona. The cell preferably is a yeast or a filamentous fungus capable of anaerobic alcoholic fermentation. The cells may further comprise one or more genetic modifications that increase the flux of the pentose phosphate pathway or, reduce unspecific aldose reductase activity. The cell preferably has the ability to produce a fermentation product such as ethanol, lactic acid, 3-hydroxy-propionic acid, &bgr;-lactam antibiotics and cephalosporins. Also provided are processes for producing these fermentation products from xylose.