  A J{\o}rgensen group is a non-elementary Kleinian group that can be generated
by two elements for which equality holds in J{\o}rgensen's Inequality. This
paper shows that the only torsion-free J{\o}rgensen group is the figure-eight
knot group, identifies all non-cocompact arithmetic J{\o}rgensen groups, and
establishes a characterization of cocompact arithmetic J{\o}rgensen groups. The
paper also defines and computes the J{\o}rgensen number of several
non-cocompact Kleinian groups including some two-bridge knot and link groups.
