  We consider actions of automorphism groups of free groups by semisimple
isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen
generators of Aut$(F_n)$ has a fixed point. If $n=3$ then either each of the
Nielsen generators has a fixed point, or else they are hyperbolic and each
Nielsen-generated $\Z^4\subset Aut(F_3)$ leaves invariant an isometrically
embedded copy of Euclidean 3-space on which it acts as a discrete group of
translations with the rhombic dodecahedron as a fundamental domain. An
abundance of actions of the second kind is described.
  Constraints on maps from Aut$(F_n)$ to mapping class groups and linear groups
are obtained. If $n\ge 2$ then neither Aut$(F_n)$ nor Out$(F_n)$ is the
fundamental group of a compact K\"ahler manifold.
