A method for making an aqueous emulsion, which method comprises: a) oligomerizing a mixture of free-radically polymerizable monomers in aqueous solution, the mixture including 1 to 45 wt. % acid-functional free-radically polymerizable comonomer(s), 0.5 to 20 wt. % adhesion-promoting free-radically polymerizable comonomer(s) comprising an aldehyde, ketone, ester, hydroxyl, silane or ureido group, and 98.5 to 50 wt. % other free-radically polymerizable comonomer(s), to form a water-soluble oligomer having a number average molecular weight (Mn) of 500 to 50,000 and a glass transition temperature (Tg) of at least 70°C, the acid-functional comonomer(s) rendering the oligomer water-soluble per se or by neutralization and the adhesion-promoting comonomer(s) imparting improved adhesion to a wood, plastic, metal or cementitious substrate when a coating composition containing the aqueous emulsion is applied and dried on such substrate, and b) emulsion polymerizing, in the presence of an aqueous solution of the oligomer, at least one free-radically polymerizable monomer and 0.1 to 20 wt. %, based on the weight of free-radically polymerizable monomer(s), of at least one di- or higher-functional chain-extending monomer(s), to form an aqueous emulsion of a hydrophobic polymer having a Tg at least 40°C lower than the Tg of the oligomer, wherein the weight of oligomer is 5 to 50 wt. % of the weight of oligomer plus hydrophobic polymer, and wherein the aqueous emulsion contains less than 0.01% free adipic dihydrazide based on the total weight of the aqueous emulsion, has a minimum film forming temperature ≤ 55°C and on drying has a Koenig hardness of at least 40 sec, wherein the Tg is measured by using differential scanning calorimetry, and wherein the Koenig hardness is measured in accordance to ISO 1522:2006. a) oligomerizing a mixture of free-radically polymerizable monomers in aqueous solution, the mixture including 1 to 45 wt. % acid-functional free-radically polymerizable comonomer(s), 0.5 to 20 wt. % adhesion-promoting free-radically polymerizable comonomer(s) comprising an aldehyde, ketone, ester, hydroxyl, silane or ureido group, and 98.5 to 50 wt. % other free-radically polymerizable comonomer(s), to form a water-soluble oligomer having a number average molecular weight (Mn) of 500 to 50,000 and a glass transition temperature (Tg) of at least 70°C, the acid-functional comonomer(s) rendering the oligomer water-soluble per se or by neutralization and the adhesion-promoting comonomer(s) imparting improved adhesion to a wood, plastic, metal or cementitious substrate when a coating composition containing the aqueous emulsion is applied and dried on such substrate, and 1 to 45 wt. % acid-functional free-radically polymerizable comonomer(s), 0.5 to 20 wt. % adhesion-promoting free-radically polymerizable comonomer(s) comprising an aldehyde, ketone, ester, hydroxyl, silane or ureido group, and 98.5 to 50 wt. % other free-radically polymerizable comonomer(s), to form a water-soluble oligomer having a number average molecular weight (Mn) of 500 to 50,000 and a glass transition temperature (Tg) of at least 70°C, the acid-functional comonomer(s) rendering the oligomer water-soluble per se or by neutralization and the adhesion-promoting comonomer(s) imparting improved adhesion to a wood, plastic, metal or cementitious substrate when a coating composition containing the aqueous emulsion is applied and dried on such substrate, and b) emulsion polymerizing, in the presence of an aqueous solution of the oligomer, at least one free-radically polymerizable monomer and 0.1 to 20 wt. %, based on the weight of free-radically polymerizable monomer(s), of at least one di- or higher-functional chain-extending monomer(s), to form an aqueous emulsion of a hydrophobic polymer having a Tg at least 40°C lower than the Tg of the oligomer, wherein the weight of oligomer is 5 to 50 wt. % of the weight of oligomer plus hydrophobic polymer, and wherein the aqueous emulsion contains less than 0.01% free adipic dihydrazide based on the total weight of the aqueous emulsion, has a minimum film forming temperature ≤ 55°C and on drying has a Koenig hardness of at least 40 sec, in the presence of an aqueous solution of the oligomer, at least one free-radically polymerizable monomer and 0.1 to 20 wt. %, based on the weight of free-radically polymerizable monomer(s), of at least one di- or higher-functional chain-extending monomer(s), to form an aqueous emulsion of a hydrophobic polymer having a Tg at least 40°C lower than the Tg of the oligomer, wherein the weight of oligomer is 5 to 50 wt. % of the weight of oligomer plus hydrophobic polymer, and wherein the aqueous emulsion contains less than 0.01% free adipic dihydrazide based on the total weight of the aqueous emulsion, has a minimum film forming temperature ≤ 55°C and on drying has a Koenig hardness of at least 40 sec,