A Willmore surface in the 3-dimensional Euclidean space is a critical point of the
square norm of the mean curvature of the surface.
The round spheres, the Clifford torus and the minimal surfaces are Willmore. For a
graph to satisfy the Willmore surface equation, its defining function is governed by
a fourth order non-linear elliptic equation. A classical theorem of Bernstein says
that an entire minimal graph must be a plane. We ask what happens to the entire
Willmore graphs. In this talk, I will discuss joint work with Tobias Lamm on the
finite energy case and with Yuxiang Li on the radially symmetric case.