Abstract of the Disclosure: Catalyst systems for the polymerization of C.sub.2 -C.sub.10 -alk-1-enes contain, as active components,a) a metallocene complex of the general formula I ##STR1## where M is titanium, zirconium, hafnium, vanadium, niobium or tantalum, X is halogen or C.sub.1 -C.sub.8 -alkyl, Y is carbon, phosphorus, sulfur, silicon or germanium, Z is C.sub.1 -C.sub.8 -alkyl, C.sub.3 -C.sub.10 -cycloalkyl or C.sub.6 -C.sub.10 -aryl, R.sup.1 and R.sup.2 are identical or different and are each C.sub.1 -C.sub.4 -alkyl, R.sup.3 to R.sup.6 are identical or different and are each hydrogen or C.sub.1 -C.sub.8 -alkyl, or two adjacent radicals R.sup.3 and R.sup.4 and R.sup.5 and R.sup.6 in each case together form a hydrocarbon ring system of 4 to 15 carbon atoms and n is 0, 1 or 2, andb) an open-chain or cyclic alumoxane compound of the general formula II or III ##STR2## where R.sup.7 is C.sub.1 -C.sub.4 -alkyl and n is from 5 to 30. The novel catalyst systems are particularly suitable for the preparation of polyalk-1-enes having high molecular weights.