In this expository article I will review some constructions of bi-invariant metrics on the contactomorphism group, and how these metrics are related to some other global rigidity phenomena in contact topology which have been discovered in the last few years, in particular the notion of orderability (due to Eliashberg and Polterovich) and an analogue in contact topology (due to Eliashberg, Kim and Poltorovich) of Gromov's symplectic non-squeezing theorem..
Source: wiktionary