  We study actions of Lie supergroups, in particular, the hitherto elusive
notion of orbits through odd (or more general) points. Following categorical
principles, we derive a conceptual framework for their treatment and therein
prove general existence theorems for the isotropy (or stabiliser) supergroups
and orbits through general points. In this setting, we show that the coadjoint
orbits always admit a (relative) supersymplectic structure of
Kirillov-Kostant-Souriau type. Applying a family version of Kirillov's orbit
method, we decompose the regular representation of an odd Abelian supergroup
into an odd direct integral of characters and construct universal families of
representations, parametrised by a supermanifold, for two different super
variants of the Heisenberg group.
