The present invention relates to a process for isomerizing reactant allyl alcohols of the formula (I) to product allyl alcohols of the general formula (II) ##STR00001## where R1 to R5 are each hydrogen or a mono- or polyunsaturated or saturated C.sub.1 C.sub.12-alkyl radical, which can be substituted or unsubstituted, in both directions of the equilibrium, which comprises isomerizing in the presence of a dioxotungsten(VI) complex of the general formula (III), ##STR00002## where L.sub.1, L.sub.2 are each independently a ligand selected from the group consisting of the aminoalcohols, the aminophenols and mixtures thereof, m, n are each 1 or 2, new dioxotungsten(VI) complexes, and also their use.