This is an introductory talk. A tree is a connected graph with no loops. An R-tree is a generalization that allows every point to be a vertex. If T is a tree or R-tree there is a group Aut(T) of automorphisms of T. Automorphisms are of two kinds depending on whether or not there is a fixed point. This gives a geometry (T,Aut(T)). One may study a homomorphism from a given group G into Aut(T). I will describe some applications to topology and group theory.