Abstract: Preparation of high-reactivity isobutene homo- or copolymers with a content of terminal vinylidene double bonds per polyisobutene chain end of at least 50 mol % and a polydispersity of preferably 1.05 to less than 3.5, by polymerizing isobutene or an isobutene-comprising monomer mixture in the presence of an aluminum trihalide-donor complex effective as a polymerization catalyst or of an alkylaluminum halide-donor complex, especially of an aluminum trichloride-donor complex, said complex comprising, as the donor, an organic compound with at least one ether function or a carboxylic ester function.