In these lectures, we will report on some properties of the space of actions of a left-orderable group on the real line. We will notably describe the almost-periodic actions, the harmonic actions and their spaces.

Irreducible lattices in semi-simple Lie groups of higher rank are not left-orderable I'll report on the problem of the left orderability of lattices in semi-simple Lie groups, and give some insight of our joint proof with Bertrand Deroin that in rank at least two, an irreducible lattice is not left-orderable. The proof will make use of the tools developed in the minicourse of Bertrand.