The outer automorphism group of the free group Out(Fn) is one of the most important and basic groups in combinatorial and geometric group theory.It acts on the outer space of metric graphs.This action is an analogy of the action of arithmetic groups on symmetric spaces,and the action of the mapping class group on the Teichmuller spaces,and has played a fundamental role in understanding Out(Fn).Both symmetric spaces and Teichmuller spaces admit many complete metrics which are invariant under the action and they are important for many applications.A longstanding open problem is to construct complete metrics on the outer space which are invariant under the action of Out(Fn).