A method for the cationic polymerization of unsaturated biological oils (e.g., vegetable oils and animal oils) based on the cationic reaction of double bonds initiated by superacids is provided. The process occurs under very mild reaction conditions (about 70-110.degree. C. and atmospheric pressure) and with short reaction times. The polymerized oils have a viscosity about 10 to 200 times higher than the initial oil and relatively high unsaturation (only about 10-30% lower than that of initial oils).