A process is described for preparing polyisobutene having a content of terminal vinylidene groups of at least 75 mol % by polymerizing isobutene or isobutenic hydrocarbon mixtures in the liquid phase in the presence of a boron trifluoride complex catalyst of the composition (BF.sub.3).sub.a.L.sup.1.sub.b.L.sup.2.sub.c.L.sup.3.sub.d where L.sup.1 is water, a primary C.sub.1 C.sub.5-alkanol and/or a secondary C.sub.3 C.sub.5-alkanol, L.sup.2 is at least one aldehyde and/or one ketone, L.sup.3 is an ether having at least 5 carbon atoms, a secondary alkanol having at least 6 carbon atoms, a primary alkanol having at least 6 carbon atoms and/or a tertiary alkanol, the b:a ratio is in the range from 0.9 to 3.0, the c:a ratio is in the range from 0.01 to 0.5, and the d:a ratio is in the range from 0 to 1.0.