The use of compounds of the formula (I) R1aRebOc•Ld  (I), in which a=zero or an integer from 0 to 6 b=an integer from 1 to 4 c=an integer from 1 to 12 d=an integer from 0 to 4 L=Lewis base and the total of a, b and c is such as to comply with the penta- or hepta-valency of rhenium, with the proviso that c is not greater than 3•b, and in which R1 is absent, identical or different and is an aliphatic hydrocarbon radical having 1 to 10 carbon atoms, an aromatic hydrocarbon radical having 6 to 10 carbon atoms or an arylalkyl radical having 7 to 9 carbon atoms, where the R1 radicals can, where appropriate, be substituted identically or differently, independently of one another, as catalysts for the selective oxidation of olefins with cleavage of C═C bonds to give the corresponding carbonyl compounds in the presence of a peroxide-containing compound, where the amount of substance ratio of olefin to peroxide-containing compound is in a range from 1:2 to 1:14.