An azetidinium-containing copolymer comprising: azetidinium-containing monomeric units derived from at least one azetidinium-containing vinylic monomer of formula in which: R" is hydrogen or methyl; p1, p2, and p3 independent of one another are zero or 1; Y 4 is a linkage selected from the group consisting of a direct bond, -O-,-NR'-, -C -NR'-, -NR'-C -, -O-C -NH-, -NH-C -O-, -NR'-C -NH-, -NH-C -NR'-, -C -O-, -O-C -, R' is hydrogen, a C 1 -C 20 unsubstituted or substituted, linear or branched alkyl group; Z 4, is a direct bond, a C 1 -C 20 unsubstituted or substituted, linear or branched alkylene divalent radical optionally containing therein one or more linkages of -O-, -NR'-, and -C -, a C 1 -C 7 alkyleneoxy C 1 -C 7 alkylene divalent radical, or a divalent radical of -(CH(R")CH 2 O) r1 -CH(R")CH 2 - in which R" is as defined above and r1 is an integer of 1 to 20; and Z 5 is a C 1 -C 20 unsubstituted or substituted, linear or branched alkyl group,-(CH 2) r2 -O-(CH 2 CH 2 O) r1 -Z 6 in which r1 is as defined above, r2 is zero or an integer of 1 to 7, and Z 6 is a C 1 -C 5 alkyl monomeric units derived from at least one vinylic monomer selected from the group consisting of a carboxyl-containing vinylic monomer, an amino-containing vinylic monomer, and combination thereof, wherein the carboxyl-containing vinylic monomer is selected from the group consisting of acrylic acid, a C 1 -C 4 -alkyl acrylic acid, N-2-acrylamidoglycolic acid, and combination thereof, wherein the amino-containing vinylic monomer is selected from the group consisting of amino-C 2 -C 4 alkyl (meth)acrylate, allylamine, vinylamine, amino-C 1 -C 4 alkyl (meth)acrylamide, N-allyl C 1 -C 12 alkanamine, a coupling reaction product of an epoxy compound having one sole epoxy group with allylamine, vinylamine, amino-C 2 -C 6 alkyl (meth)acrylate, or amino-C 2 -C 6 alkyl (meth)acrylamide, a coupling reaction product of an C 1 -C 12 alkanamine or C 2 -C 12 aminoalkanol or with an epoxy-containing vinylic monomer, and combinations thereof. azetidinium-containing monomeric units derived from at least one azetidinium-containing vinylic monomer of formula in which: R" is hydrogen or methyl; p1, p2, and p3 independent of one another are zero or 1; Y 4 is a linkage selected from the group consisting of a direct bond, -O-,-NR'-, -C -NR'-, -NR'-C -, -O-C -NH-, -NH-C -O-, -NR'-C -NH-, -NH-C -NR'-, -C -O-, -O-C -, R' is hydrogen, a C 1 -C 20 unsubstituted or substituted, linear or branched alkyl group; Z 4, is a direct bond, a C 1 -C 20 unsubstituted or substituted, linear or branched alkylene divalent radical optionally containing therein one or more linkages of -O-, -NR'-, and -C -, a C 1 -C 7 alkyleneoxy C 1 -C 7 alkylene divalent radical, or a divalent radical of -(CH(R")CH 2 O) r1 -CH(R")CH 2 - in which R" is as defined above and r1 is an integer of 1 to 20; and Z 5 is a C 1 -C 20 unsubstituted or substituted, linear or branched alkyl group,-(CH 2) r2 -O-(CH 2 CH 2 O) r1 -Z 6 in which r1 is as defined above, r2 is zero or an integer of 1 to 7, and Z 6 is a C 1 -C 5 alkyl R" is hydrogen or methyl; p1, p2, and p3 independent of one another are zero or 1; Y 4 is a linkage selected from the group consisting of a direct bond, -O-,-NR'-, -C -NR'-, -NR'-C -, -O-C -NH-, -NH-C -O-, -NR'-C -NH-, -NH-C -NR'-, -C -O-, -O-C -, R' is hydrogen, a C 1 -C 20 unsubstituted or substituted, linear or branched alkyl group; Z 4, is a direct bond, a C 1 -C 20 unsubstituted or substituted, linear or branched alkylene divalent radical optionally containing therein one or more linkages of -O-, -NR'-, and -C -, a C 1 -C 7 alkyleneoxy C 1 -C 7 alkylene divalent radical, or a divalent radical of -(CH(R")CH 2 O) r1 -CH(R")CH 2 - in which R" is as defined above and r1 is an integer of 1 to 20; and Z 5 is a C 1 -C 20 unsubstituted or substituted, linear or branched alkyl group,-(CH 2) r2 -O-(CH 2 CH 2 O) r1 -Z 6 in which r1 is as defined above, r2 is zero or an integer of 1 to 7, and Z 6 is a C 1 -C 5 alkyl monomeric units derived from at least one vinylic monomer selected from the group consisting of a carboxyl-containing vinylic monomer, an amino-containing vinylic monomer, and combination thereof, wherein the carboxyl-containing vinylic monomer is selected from the group consisting of acrylic acid, a C 1 -C 4 -alkyl acrylic acid, N-2-acrylamidoglycolic acid, and combination thereof, wherein the amino-containing vinylic monomer is selected from the group consisting of amino-C 2 -C 4 alkyl (meth)acrylate, allylamine, vinylamine, amino-C 1 -C 4 alkyl (meth)acrylamide, N-allyl C 1 -C 12 alkanamine, a coupling reaction product of an epoxy compound having one sole epoxy group with allylamine, vinylamine, amino-C 2 -C 6 alkyl (meth)acrylate, or amino-C 2 -C 6 alkyl (meth)acrylamide, a coupling reaction product of an C 1 -C 12 alkanamine or C 2 -C 12 aminoalkanol or with an epoxy-containing vinylic monomer, and combinations thereof.