In this talk I will give some general background on the field of exact enumeration of tilings and perfect matchings, pointing out a few classical theorems and some very recent results. Then I will focus on presenting the proof for one such result, concerning some of Propp's conjectures on the number of tilings of certain regions called benzels. To do this, I will recall an older result of mine (the matchings factorization theorem for symmetric graphs), and present a recent extension of it that is crucial for the proof. Some other consequences of this extension, including a surprising semi-factorization for lozenge tilings of certain nearly-symmetric hexagonal regions with holes, will also be presented. This is joint work with Seok Hyun Byun and Yi-Lin Lee.