  We characterize hyperfinite bipartite graphings that admit measurable perfect
matchings. In particular, we prove that every regular hyperfinite bipartite
graphing admits a measurable perfect matching if it is one-ended or the degree
is odd. We give several applications of this result, answering various open
questions in the field. For instance, we extend the Lyons--Nazarov theorem by
characterizing bipartite Cayley graphs which admit a factor of iid perfect
matching, answering the bipartite case of a well-known question of Lyons and
Nazarov, popularized by Kechris and Marks. Moreover, we show how our results
apply to measurable equidecompositions and, in particular, generalize the
recent result of Grabowski, M\'ath\'e and Pikhurko on the measurable circle
squaring. Our approach applies more generally to rounding measurable perfect
fractional matchings.
