  Contact manifolds are odd-dimensional smooth manifolds endowed with a
maximally non-integrable field of hyperplanes. They are intimately related to
symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a
closed non-degenerate 2-form. Although in symplectic topology a famous
bi-invariant metric, the Hofer metric, has been studied since more than 20
years ago, it is only recently that some somehow analogous bi-invariant metrics
have been discovered on the group of diffeomorphisms that preserve a contact
structure. 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.
