The catalyst comprises at least one alkali metal on a support which has the general formula (I) A.sub.n' Ti.sub.m' Zr.sub.p' Hf.sub.q' O.sub.n'+2(m'+p'+q') (I) where A is a divalent metal, 20.multidot.(m'+p'+q')>n'>0.05.multidot.(m'+p'+q'), and which may be doped with at least one compound of an alkali metal and/or alkaline earth metal, where the alkali metal/support ratio by weight is 0.01-5:1 and, when a dopant is present, the dopant/support ratio by weight is 0.01-5:1, and where the proportion of support phase corresponding to the ZrO.sub.2 structure or an alkaline earth metal oxide structure or consisting of ZrO.sub.2 or alkaline earth metal oxide is less than 10% by weight. The catalyst is used for the side-chain alkylation or alkenylation of alkylaromatic compounds with olefins or diolefins, for the double-bond isomerization of olefins, for the dimerization of olefins, for the coupling of aromatic nuclei and for the amination of olefins and conjugated diolefins.