  A non-Markovian stochastic Schroedinger equation for a quantum system coupled
to an environment of harmonic oscillators is presented. Its solutions, when
averaged over the noise, reproduce the standard reduced density operator
without any approximation. We illustrate the power of this approach with
several examples, including exponentially decaying bath correlations and
extreme non-Markovian cases, where the `environment' consists of only a single
oscillator. The latter case shows the decay and revival of a `Schroedinger cat'
state. For strong coupling to a dissipative environment with memory, the
asymptotic state can be reached in a finite time. Our description of open
systems is compatible with different positions of the `Heisenberg cut' between
system and environment.
