  We present a nonlinear stochastic Schroedinger equation for pure states
describing non-Markovian diffusion of quantum trajectories. It provides an
unravelling of the evolution of a quantum system coupled to a finite or
infinite number of harmonic oscillators, without any approximation. Its power
is illustrated by several examples, including measurement-like situations,
dissipation, and quantum Brownian motion. In some examples, we treat the
environment phenomenologically as an infinite reservoir with fluctuations of
arbitrary correlation. In other examples the environment consists of a finite
number of oscillators. In these quasi-periodic cases we see the reversible
decay of a `Schroedinger cat' state. Finally, our description of open systems
is compatible with different positions of the `Heisenberg cut' between system
and environment.
