We go through the history of measurable perfect matchings from the Banach-Tarski paradox via circle squaring and report on recent progress. We construct a dd-regular treeing (for every d>2d>2) without a measurable perfect matching. We show that the Hall condition is essentially sufficient in the hyperfinite, one-ended, bipartite case. This allows us to characterize bipartite Cayley graphs with factor of i.i.d. perfect matchings extending the Lyons-Nazarov theorem. We apply these to Gardner's conjecture for uniformly distributed sets, to balanced orientations, and to get new, simple proofs of the measurable circle squaring. We prove the analogous theorems in the context of rounding flows, too. Partially joint work with Matt Bowen and Marcin Sabok.