Star-shaped polymers (4) that consist of poly(p-alkoxystyrene) arms and a microgel core of a divinyl compound (1b) were synthesized in high yield (>95%) on the basis of the living cationic polymerization with the HI/ZnI2 initiating system. For example, the polymerization of p-methoxystyrene (pMOS) with HI/ZnI2 at -15 degrees C in methylene chloride led to a living polymer, which was subsequently allowed to react with a small amount of the linking agent 1b [CH2=CHC6H4O(CH2)(3)OC6H4CH=CH2] to give star-shaped polymers 4, soluble in common organic solvents including chloroform, methylene chloride, and THF. The M(w) (by light scattering) of 4 ranged from 5 x 10(4) to 60 x 10(4), and the number of arms from 7 to 50 per molecule, which increased with increasing [1b](0)/[living end] ratio. Similar results were obtained with p-tert-butoxystyrene having a bulkier substituent than pMOS. In contrast to the bis(alkoxystyrene)-type compound Ib, the use of a bifunctional vinyl ether [1a; CH2=CHOCH2CH2-OC6H4C(CH3)(2)C6H4OCH2CH2OCH=CH2] as a linking agent resulted in a sharp decrease in the yield of the star-shaped polymers.