The invention provides a process for making acetic anhydride, wherein methyl acetate and/or dimethylether is (are) reacted with carbon monoxide under substantially anhydrous conditions, at temperatures of 350 to 575 K and under pressures of 1 to 300 bars in the presence of a catalyst system comprised of noble metals belonging to group VIII of the periodic system of the elements, or their compounds and iodine and/or its compounds. More specifically, an aliphatic carboxylic acid with 1 to 8 carbon atoms and at least one heterocyclic aromatic compound, in which at least one hetero atom is a quaternary nitrogen atom are used as promoters.