We will present a notion of limit of permutations,recently introduced by Hoppen et al.
The limiting objects, called permutons, describe the scaling limit of large random permutations.

We will consider various probability distributions on the set
of permutations of a given size
and present convergence results under these distributions in the sense of permutons (when the size tends to infinity).
In some family of examples, the limiting permutons are connected to the Brownian (or stable) excursions.