Abstract: Biharmonic and conformal biharmonic maps between Riemannian manifolds are two fourth order generalizations of the well-studied notion of harmonic maps. Since both equations arise as critical points of an energy functional which is non-coercive and due to the large number of derivatives it is a challenging task to establish existence results for the latter. After providing a brief introduction to biharmonic and conformal biharmonic maps we will focus on the case of maps to the Euclidean sphere. We will present a number of recent existence results for both equations and point out the influence of the domain manifold on the existence of non-trivial solutions.