Olefins are hydroformylated to give alcohols and/or aldehydes in a plurality of hydroformylation stages, each of which comprises: a) hydroformylating olefins having a carbon atom content of 6 to 24 carbon atoms in the presence of a cobalt- or rhodium catalyst in a reactor to the point of conversion of olefin reactant to product of 20 to 98%; b) removing the catalyst from the resulting liquid discharged from the reactor; c) separating the resulting liquid hydroformylation mixture into a low-boiler fraction comprising olefins and paraffins, and a bottoms fraction comprising aldehydes and/or alcohols; and d) reacting the olefins present in the low-boiler fraction in subsequent process stages comprising steps a, b and c and combining the bottoms fractions of process steps c) of all process stages.