Water-soluble copolymers containing copolymerized vinylamine units are prepared by copolymerizing (a) from 95 to 10 mol % of N-vinylformamide and (b) from 5 to 90 mol % of an ethylenically unsaturated monomer from the group consisting of vinyl acetate, vinyl propionate, C.sub.1 -C.sub.4 -alkyl vinyl ethers, the esters, nitriles and amines of acrylic acid and methacrylic acid and N-vinylpyrrolidone, and then eliminating from 30 to 100 mol % of the formyl groups from the copolymer. Preferred copolymers are copolymers of N-vinyl-formamide and vinyl acetate, in which from 30 to 100 mol % of the monomer units are hydrolyzed. The copolymers are used in papermaking to increase the dry strength and wet strength of the paper.