Tertiary hydroxyalkylxanthines having the general formula (I), characterized by the fact that at least one of the residues R.sup.1 and R.sup.3 represents a tertiary hydroxyalkyl group of formula (Ia) in which R.sup.4 represents an alkyl group having up to 3 carbon atoms and n a whole number from 2 to 5; the other residue which may remain R.sup.1 or R.sup.3 is a hydrogen atom or an aliphatic hydrocarbon residue R.sup.5 having up to 6 carbon atoms, in which the carbon chain may be separated by a maximum of two oxygen atoms or be substituted by an oxo group or two hydroxy groups at the most, and R.sup.2 represents an alkyl group with 1 to 4 carbon atoms. Also described is a process for manufacturing these compounds, which are suitable for the manufacture of medicines, in particular for use in the prevention and/or treatment of peripheral and/or cerebral irrigation disorders. ##STR1##