  Given a spacelike foliation of a spacetime and a marginally outer trapped
surface S on some initial leaf, we prove that under a suitable stability
condition S is contained in a ``horizon'', i.e. a smooth 3-surface foliated by
marginally outer trapped slices which lie in the leaves of the given foliation.
We also show that under rather weak energy conditions this horizon must be
either achronal or spacelike everywhere. Furthermore, we discuss the relation
between ``bounding'' and ``stability'' properties of marginally outer trapped
surfaces.
