  Motivated by the rich theory of harmonic maps from a 2-sphere, we study
biharmonic maps from a 2-sphere in this paper. We first derive biharmonic
equation for rotationally symmetric maps between rotationally symmetric
2-manifolds. We then apply the equation to obtain a classification of
biharmonic maps in a family of rotationally symmetric maps between 2-spheres.
We also find many examples of proper biharmonic maps defined locally on a
2-sphere. Our results seem to suggest that any biharmonic map
$S^2\longrightarrow (N^n, h)$ be a weakly conformal immersion.
