Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. In two recent papers --- with Chan, Dipierro and Valdinoci, and with Caselli and Florit--- we set the ground for a new approximation based on nonlocal minimal surfaces. In the first paper, we prove that stable s-minimal surfaces in the unit ball of R^3 satisfy curvature estimates that are robust as s approaches 1 (i.e. as the energy approaches that of classical minimal surfaces). Moreover, we obtain optimal sheet separation estimates and show that critical interactions are encoded by nontrivial solutions to a (local) "Toda type" system. As a nontrivial application, we establish that hyperplanes are the only stable s-minimal hypersurfaces in R^4, for s sufficiently close to 1. In the second paper, we establish the existence of infinitely many nonlocal minimal surfaces in every closed manifold (i.e., a version of Yau's conjecture).