  We propose a new notion called \emph{infinity-harmonic maps}between
Riemannain manifolds. These are natural generalizations of the well known
notion of infinity harmonic functions and are also the limiting case of $p$%
-harmonic maps as $p\to \infty $.
  Infinity harmoncity appears in many familiar contexts. For example, metric
projection onto the orbit of an isometric group action from a tubular
neighborhood is infinity harmonic.
  Unfortunately, infinity-harmonicity is not preserved under composition. Those
infinity harmonic maps that always preserve infinity harmonicity under pull
back are called infinity harmonic morphisms. We show that infinity harmonic
morphisms are precisely horizontally homothetic mas.
  Many example of infinity-harmonic maps are given, including some very
important and well-known classes of maps between Riemannian manifolds.
