  $\infty$-Harmonic maps are a generalization of $\infty$-harmonic functions.
They can be viewed as the limiting cases of p-harmonic maps as p goes to
infinity. In this paper, we give complete classifications of linear and
quadratic $\infty$-harmonic maps from and into a sphere, quadratic
$\infty$-harmonic maps between Euclidean spaces. We describe all linear and
quadratic $\infty$-harmonic maps between Nil and Euclidean spaces, between Sol
and Euclidean spaces. We also study holomorphic $\infty$-harmonic maps between
complex Euclidean spaces.
