Process for the production of hydrogenated, ring-opened metathesis (co)polymers from cyclic olefins using the ruthenium-carbene complexes of the general formula (I) ##STR1## in which R.sub.1 and R.sub.2 mutually independently mean hydrogen, C.sub.2 -C.sub.20 alkenyl, C.sub.1 -C.sub.20 alkyl, aryl, C.sub.1 -C.sub.20 carboxylate, C.sub.1 -C.sub.20 alkoxy, C.sub.2 -C.sub.20 alkenyloxy, aryloxy, C.sub.2 -C.sub.20 alkoxycarbon-yl,C.sub.1 -C.sub.20 alkylthio, which may optionally be substituted by C.sub.1 -C.sub.5 alkyl, halogen, C.sub.1 -C.sub.5 alkoxy or by phenyl optionally substituted by C.sub.1 -C.sub.5 alkyl, halogen, C.sub.1 -C.sub.5 alkoxy;X.sub.1 and X.sub.2 mutually independently mean any desired anionic ligand;L.sub.1 and L.sub.2 mutually independently mean any desired neutral electron donorand two or three of X.sub.1, X.sub.2, L.sub.1 and L.sub.2 may furthermore together form a multi-dentate chelating ligand,as catalysts for the polymerisation stage, wherein the catalyst is optionally modified after polymerisation by adding a modifier and the polymerisation product is hydrogenated as a solution (optionally diluted with the same inert solvent as during polymerisation or with another inert solvent) without addition of an extra hydrogenation catalyst at temperatures of 0.degree. C. to 200.degree. C. preferably of 35.degree. C. to 150.degree. C. under a hydrogen pressure of 2 to 200 bar, preferably of 10 to 50 bar, wherein at least 60%, preferably at least 95% of the olefinic double bonds are saturated.