  We derive a hierarchy of stochastic evolution equations for pure states
(quantum trajectories) to efficiently solve open quantum system dynamics with
non-Markovian structured environments. From this hierarchy of pure states
(HOPS) the exact reduced density operator is obtained as an ensemble average.
We demonstrate the power of HOPS by applying it to the Spin-Boson model, the
calculation of absorption spectra of molecular aggregates and energy transfer
in a photosynthetic pigment-protein complex.
