A Hausdorﬀ measure version of the Duﬃn-Schaeﬀer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of Rk to Hausdorﬀ measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorﬀ theory. This is rather surprising since the latter theory is viewed to be a subtle reﬁnement of the former. .