  Biharmonic maps between surfaces are studied in this paper. We compute the
bitension field of a map between surfaces with conformal metrics in complex
coordinates. As applications, we show that a linear map from Euclidean plane
into $(\mathbb{R}^2, \sigma^2dwd\bar w)$ is always biharmonic if the conformal
factor $\sigma$ is bi-analytic; we construct a family of such $ \sigma$, and we
give a classification of linear biharmonic maps between $2$-spheres. We also
study biharmonic maps between surfaces with warped product metrics. This
includes a classification of linear biharmonic maps between hyperbolic planes
and some constructions of many proper biharmonic maps into a circular cone or a
helicoid.
