To obtain the subject polymer excellent in wet brakeage and fuel economy without substantially dropping its abrasion resistance by designing itself to include isobutylene repeating unit and introduce specific free radical into the molecule.

This polymer is obtained by introducing one or more kinds of free radical stable at normal temperatures and in the presence of oxygen such as nitroxide radical, hydrazyl radical, aryloxy radical and trityl radical into the molecule, pref. onto the molecular terminals, of a polymer comprising isobutylene repeating unit; wherein the polymer comprising isobutylene repeating unit is e.g. polyisobutylene, polybutene, butyl rubber, brominated butyl rubber, isobutylene-maleic anhydride copolymer, isobutylene-p-methylstyrene-p- bromomethylstyrene copolymer. The other rubber composition is obtained by including the polymer containing the above free radical.