In this talk I will review some recent progress in the understanding of the optimal matching problem. While the work of Ajtai-Komlos-Tusnady in the 80's on this classical optimization problem attracted a lot of attention from the probability community (see the book by Talagrand), this problem has seen a renewed interest from the PDE community thanks to the ansatz proposed by Caracciolo Lucibello, Parisi and Sicuro in 2014. I will explain to which extent this ansatz can be rigorously justified and show how it leads to a deeper understanding of this problem.