The belief propagation method, also known as the cavity method in physics, has found many applications in discrete optimisation, statistical mechanics and computer science since its genesis about half a century ago. One of the first rigorous mathematical treatments of this technique dates back to the beginning of the millennium with the foundational work by Aldous on the random assignment problem. In this talk, I will focus on matching problems/monomer-dimer ground states and give an overview of the method and its limitations. I will also discuss some recent and ongoing progress using belief propagation towards understanding the local law of disordered monomer-dimer ground states on trees.