The concept of quasiconformal motions was introduced in a paper of Sullivan and Thurston. In this talk, we will show that a continuous motion of a finite set $E$ in the Riemann sphere over a simply connected parameter space $V$ can be extended to a quasiconformal motion of the sphere over $V$, and that such an extension is unique up to isotopy rel $E$. This fact leads to results on continuity for the extremal length of a general family of curves, and in particular for modulus of annuli and lengths of closed Poincare geodesics. We will discuss these geometrical applications. This is a joint work with Yunping Jiang.