  The ancient unsolved problem of congruent numbers has been reduced to one of
the major questions of contemporary arithmetic: the finiteness of the number of
curves over $\bf Q$ which become isomorphic at every place to a given curve. We
give an elementary introduction to congruent numbers and their conjectural
characterisation, discuss local-to-global issues leading to the finiteness
problem, and list a few results and conjectures in the arithmetic theory of
elliptic curves.
