2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156980The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored.31 pages, Dedicated to Klaus Roth on the occasion of his 80th birthdayNumber Theory11J83text