  The 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.
