We look at *geometric* limits of large random *non-uniform* permutations. We mainly consider two theories for limits of permutations: *permuton limits*, introduced by Hoppen, Kohayakawa, Moreira, Rath, and Sampaio to define a notion of scaling limits for permutations; and *Benjamini-Schramm limits*, introduced by the author to define a notion of local limits for permutations.

The models of random permutations that we consider are mainly *constrained models*, that is, uniform permutations belonging to a given subset of the set of all permutations. We often identify this subset using pattern-avoidance, focusing on: permutations avoiding a pattern of length three, substitution-closed classes, (almost) square permutations, permutation families encoded by generating trees, and Baxter permutations.

We explore some universal phenomena for the models mentioned above. For Benjamini-Schramm limits we explore a *concentration phenomenon* for the limiting objects. For permuton limits we deepen the study of some known universal permutons, called *biased Brownian separable permutons*, and we introduce some new ones, called *Baxter permuton* and *skew Brownian permutons*. In addition, for (almost) square permutations, we investigate the occurrence of a phase transition for the limiting permutons.

On the way, we establish various combinatorial results both for permutations and other related objects. Among others, we give a complete description of the *feasible region for consecutive patterns* as the *cycle polytope* of a specific graph; and we find new bijections relating Baxter permutations, bipolar orientations, walks in cones, and a new family of discrete objects called *coalescent-walk processes*.