A functionalized polymer having excellent adhesion, printability, and compatibility in polymer blends or the like which are obtained by introducing an epoxy group, a hydroxyl group or a sulfonic group into a syndiotactic .alpha.-olefin polymer having an olefinic unsaturated bond at its terminus is disclosed. The terminal unsaturated syndiotactic .alpha.-olefin polymer is obtained by polymerizing an .alpha.-olefin with a catalyst comprising the combination of a metallocene compound and an alumoxane, and the functional groups are obtained by the oxidation, sulfonation or the like of the terminal unsaturated group, by the addition of a compound containing the corresponding functional group to the unsaturated bond, or by the conversion of a precursor polymer containing a precursor of the functional group into a polymer containing the functional group desired.