  We present a formulation of non-Markovian quantum trajectories for open
systems from a measurement theory perspective. In our treatment there are three
distinct ways in which non-Markovian behavior can arise; a mode dependent
coupling between bath (reservoir) and system, a dispersive bath, and by
spectral detection of the output into the bath. In the first two cases the
non-Markovian behavior is intrinsic to the interaction, in the third case the
non-Markovian behavior arises from the method of detection. We focus in detail
on the trajectories which simulate real-time spectral detection of the light
emitted from a localized system. In this case, the non-Markovian behavior
arises from the uncertainty in the time of emission of particles that are later
detected. The results of computer simulations of the spectral detection of the
spontaneous emission from a strongly driven two-level atom are presented.
