Outer Space is a continuous space which paramaterizes graphs with a fixed (free) fundamental group. It was introduced in the mid 1980's to study automorphism groups of free groups. In this talk I will describe Outer Space, and explain how it is related to various chain complexes of graphs used by M. Kontsevich to study the homology of certain infinite-dimensional Lie algebras. I will also describe joint work with J. Conant, in which geometric insights gained from this perspective help to clarify some new algebraic structures which we defined on these chain complexes.