  We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\geq 1$)
and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every
free homotopy class in $[M^{2m}, N]$, provided that the homotopy group
$\pi_{2m}(N)$ is trivial. This generalizes the celebrated existence results for
harmonic maps and biharmonic maps. We also prove that there exists a
non-constant smooth polyharmonic map from $\mathbb{R}^{2m}$ to $N$ by a blowup
analysis at an energy-concentration point for an energy-minimizing sequence if
the convergence fails to be strong.
