  By starting from the stochastic Schr\"odinger equation and quantum trajectory
theory, we introduce memory effects by considering stochastic adapted
coefficients. As an example of a natural non-Markovian extension of the theory
of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise
as the output driving process. Under certain conditions a random Hamiltonian
evolution is recovered. Moreover, we show that our non-Markovian stochastic
Schr\"odinger equations unravel some master equations with memory kernels.
