In this talk, we discuss recent work with Albers, Cieliebak, Fish, Frauenfelder, Hofer and Paternain on several aspects of the three body problem. The ultimate goal of this project is to use modern, holomorphic curve techniques to investigate the dynamics of the three body problem.

We shall describe how contact topology and other geometrical methods can be used to understand some aspects of the three-body problem. In particular, we shall discuss how to find global surfaces of section, a tool first developed by Poincar\'e to discretize the dynamics of a flow.