Holomorphic motions are isotopies f_z(w), holomorphic in the parameter z, of maps that are themselves (usually) not holomorphic but quasiconformal. They were introduced by Mane, Sad and Sullivan in the context of their work on the topological dynamics of rational selfmaps of the Riemann sphere. The talk will start with the outline of this background and then will discuss the solution of the problem of existence of extensions of holomorphic motion of a set to holomorphic motions of the whole Riemann sphere (posed by Sullivan and Thurston. Some applications will be sketched.