Alkylidene complexes of ruthenium containing N-heterocyclic carbene ligands and their use as highly active, selective catalysts for olefin metathesisThe invention relates to a complex of ruthenium of the structural formula I, ##STR00001## where X.sup.1 and X.sup.2 are identical or different and are each an anionic ligand, R.sup.1 and R.sup.2 are identical or different and can also contain a ring, and R.sup.1 and R.sup.2 are each hydrogen or/and a hydrocarbon group, the ligand L.sup.1 is an N-heterocyclic carbene and the ligand L.sup.2 is an uncharged electron donor, in particular an N-heterocyclic carbene or an amine, imine, phosphine, phosphite, stibine, arsine, carbonyl compound, carboxyl compound, nitrile, alcohol, ether, thiol or thioether, where R.sup.1, R.sup.2, R.sup.3 and R.sup.4 are hydrogen or/and hydrocarbon groups. The invention further relates to a process for preparing acyclic olefins having two or more carbon atoms or/and cyclic olefins having four or more carbon atoms from acyclic olefins having two or more carbon atoms or/and from cyclic olefins having four or more carbon atoms by an olefin metathesis reaction in the presence of at least one catalyst, wherein a complex is used as catalyst and R'.sup.1, R'.sup.2, R'.sup.3 and R'.sup.4 are hydrogen or/and hydrocarbon groups.