Abstract The homogeneous Ziegler-Natta catalyst, consisting of zirconocene compounds and methylalumoxane, leads to extremely high polymerization activities. With bis(cyclopentadienyl) zirconium dichloride, polyethylene with activities up to 40 x 106 g PE/g Zr·h is formed. The higher the degree of oligomerization of the alumoxane the more active are the catalysts. The density of polyethylene (LLDPE) can be lowered by using 1-butene or 1- hexene as comonomers. Incorporation of dienes gives EPDM-elastomers. Highly isotactic polypropylene can be prepared by using chiral rac-ethylene bis(tetrahydroindenyl) zirconium dichloride together with methylalumoxane as cocatalyst. The toluene soluble part here consists of less than 0.2 weight per cent. Cyclopentene gives a highly isotactic and insoluble polymer. With the R or S enantiomers of the catalyst it is also possible to produce optically active oligomers. The side reactions and the mechanism are discussed.