  Motivated by the beautiful theory and the rich applications of harmonic
conformal immersions and conformal immersions of constant mean curvature (CMC)
surfaces, we study biharmonic conformal immersions of surfaces into a generic
3-manifold. We first derive an invariant equation for such immersions, we then
try to answer the question, "what surfaces can be biharmonically conformally
immersed into Euclidean 3-space?" We prove that a circular cylinder is the only
CMC surface that can be biharmonically conformally immersed into a Euclidean
3-space; We obtain a classification of biharmonic conformal immersions of
complete CMC surfaces into a Euclidean 3-space and a hyperbolic 3-spaces. We
also study rotational surfaces that can be biharmonically conformally immersed
into a Euclidean 3-space and prove that a circular cone can never such an
immersion.
