  Conformal harmonic maps from a 4-dimensional conformal manifold to a
Riemannian manifold are maps satisfying a certain conformally invariant fourth
order equation. We prove a general existence result for conformal harmonic
maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses
a geometric flow and relies on results of Gursky-Viaclovsky and Lamm.
