Based on a microscopic model of an open system consisting of a boson field
coupled linearly to a system, we formulate quantum trajectories as continuo
us measurements of the output field from the system. We present a formulati
on for the case when the field quantity that is being monitored effectively
commutes with itself at different times. in general, this results in a non
-Markovian evolution for the reduced system conditioned on the measurements
. Under the finite-memory time approximation this evolution can be numerica
lly simulated. The generalization of the techniques of quantum trajectories
to the non-Markovian regime is expected to contribute to our understanding
of open system problems beyond the Born-Markov approximation.