This is a talk addressing to a general audience about an important and interesting topic called holomorphic motions. Holomorphic motions were defined from the study of complex dynamical systems and became an important tool in the study of complex dynamical systems and in the study of Teichmuller theory.
In this talk I will give a review about the development of holomorphic motions. I will also introduce some important applications of holomorphic motions into complex dynamical systems and into Teichmuller theory. I will also talk about the lifting problem in Teichmuller theory and our solution for this problem in the unit disk case. Finally, I will mention our recent work to connect the extendibility of holomorphic motions with homotopy and monodromy in topology.