  In this article we initiate a thorough geometric study of the conformal
bienergy functional which consists of the standard bienergy augmented by two
additional curvature terms. The conformal bienergy is conformally invariant in
dimension four and its precise structure is motivated by the Paneitz operator
from conformal geometry. The critical points of the conformal bienergy are
called conformal biharmonic maps.
  Besides establishing a number of basic results on conformal biharmonic maps,
we pay special attention to conformal biharmonic hypersurfaces in space forms.
For hypersurfaces in spheres, we determine all conformal biharmonic
hyperspheres and then we classify all conformal biharmonic generalized Clifford
tori. Moreover, in sharp contrast to biharmonic hypersurfaces, we show that
there also exist conformal biharmonic hypersurfaces of hyperbolic space,
pointing out a fundamental difference between biharmonic and conformal
biharmonic hypersurfaces.
  Finally, we also study the stability of the conformal biharmonic hyperspheres
in spheres and explicitly compute their index and nullity. In particular, we
obtain that the index of the equator $\mathbb{S}^4$ of $\mathbb{S}^5$ is zero,
i.e., it is stable, while the index of the equator $\mathbb{S}^5$ of
$\mathbb{S}^6$ is seven.
