  After having introduced the notion of universality in statistical mechanics
and its importance for our comprehension of the macroscopic behavior of
interacting systems, I review recent progress in the understanding of the
scaling limit of lattice critical models, including a quantitative
characterization of the limiting distribution and the robustness of the limit
under perturbations of the microscopic Hamiltonian. Specifically, I focus on
two classes of non-exactly-solvable two-dimensional systems: non-planar Ising
models and interacting dimers. In both settings, I describe the conjectures on
the expected structure of the scaling limit, review the progress towards their
proof, and state some of the recent results on the universality of the limit,
which I contributed to. Finally, I outline the ideas and methods involved in
the proofs, describe some of the perspectives opened by these results, and
propose several open problems.
