Abstract: The invention relates to glyceryl nucleotides of the formula Ia in which a) one of the radicals A1, A2 and A3 is a hydrogen atom or a radical selected from hydroxyl, mercapto, alkyl, alkenyl, polyoxyalkenyl, aryl, acyl, alkyloxy, alkenyloxy, polyoxyalkenyloxy, acyloxy, aryloxy, alkylthio, alkenylthio, acylthio and arylthio, where the alkyl, alkenyl and acyl radicals are optionally substituted by from 1 to 3 aryl radicals; and b1) two of the remaining radicals A1, A2 and A3 are two nucleoside groups which are different from each other; or b2) one of the remaining radicals A1, A2 and A3 is a nucleoside group and the other of the remaining radicals is a hydroxycarbonyl group, where at least one of the nucleoside groups is not a naturally occurring nucleoside group; to processes for preparing these compounds, to pharmaceutical remedies which comprise these compounds, and to the use of these compounds for treating cancer diseases and infectious diseases.