Abstract: Biharmonic maps are a fourth order generalization of the well-studied harmonic map equation. They can be characterized as critical points of the bienergy for maps between two Riemannian manifolds. While the energy for maps, whose critical points are precisely harmonic maps, is invariant under conformal transformations for a two-dimensional domain the bienergy is not invariant under conformal transformations in any dimension. In this talk we will introduce a version of the bienergy that is conformally invariant on four-dimensional manifolds and whose critical points are called conformal biharmonic maps. We will present the basic properties of conformal biharmonic maps with particular attention to conformal biharmonic hypersurfaces in space forms and the stability of conformal biharmonic hyperspheres. Moreover, we will point out many surprising differences between biharmonic and conformal biharmonic maps. This is joint work with Simona Nistor and Cezar Oniciuc.