Harmonic maps are mappings $u:M\to N$ between Riemannian manifolds which are critical points of the energy functional $E(u):=\int|Du|^2$. They are classical objects of study in Geometric Analysis. A higher order generalization is to study critical points of higher order energies like $\int|D^m u|^2$, called polyharmonic maps. They solve a critically nonlinear partial differential equation. The nonlinearity is of a special structure, maybe due to its geometric nature. This additional structure allows to prove some partial regularity results for minimizers or stationary points, but some basic regularity questions are also still open.