It is a central theme in polymer science to control the topology and architecture of polymers for the preparation of polymeric materials with new properties. However, the application of self-accelerating copper-catalyzed azide-alkyne cycloaddition (CuAAC) reaction to construct polymers was just emerged for several years.In this thesis, I developed the polymerization method by utilizing the self-accelerating CuAAC reaction with easy experimental setup, fast reaction rate, tunable external ligands addition. The CuAAC reaction was applied to construct branched polymers with different topologies including molecular brushes, star polymers and hyperbranched polymers in this body of research. More interestingly, the obtained hyperbranched polymers exhibited good performance in the catalysis of Heck reaction.