  There are good motivations for considering some type of quantum histories
formalism. Several possible formalisms are known, defined by different
definitions of event and by different selection criteria for sets of histories.
These formalisms have a natural interpretation, according to which nature
somehow chooses one set of histories from among those allowed, and then
randomly chooses to realise one history from that set; other interpretations
are possible, but their scientific implications are essentially the same.
  The selection criteria proposed to date are reasonably natural, and certainly
raise new questions. For example, the validity of ordering inferences which we
normally take for granted --- such as that a particle in one region is
necessarily in a larger region containing it --- depends on whether or not our
history respects the criterion of ordered consistency, or merely consistency.
  However, the known selection criteria, including consistency and medium
decoherence, are very weak. It is not possible to derive the predictions of
classical mechanics or Copenhagen quantum mechanics from the theories they
define, even given observational data in an extended time interval. Attempts to
refine the consistent histories approach so as to solve this problem by finding
a definition of quasiclassicality have so far not succeeded.
  On the other hand, it is shown that dynamical collapse models, of the type
originally proposed by Ghirardi-Rimini-Weber, can be re-interpreted as set
selection criteria within a quantum histories framework, in which context they
appear as candidate solutions to the set selection problem. This suggests a new
route to relativistic generalisation of these models, since covariant
definitions of a quantum event are known.
