  In recent years, a large number of papers have appeared that dealt with
$\EPEM$ pair production in heavy ion collisions at high energies. To a large
extent these studies were motivated by the existence of relativistic heavy ion
accelerators all over the world. There pair production can be studied in so
called ``ultra-peripheral collisions'', where the ions do not come close enough
to interact strongly with each other. Various different methods have been used
and it is the purpose of this review to present a unified picture of the
present status of the field.
