In all models listed so far, the noise responsible for the collapse is Markovian (memoryless): either a Poisson process in the discrete GRW model, or a white noise in the continuous models. The models can be generalized to include arbitrary (colored) noises, possibly with a frequency cutoff: the CSL model model has been extended to its colored version (cCSL), as well as the QMUPL model (cQMUPL). In these new models the collapse properties remain basically unaltered, but specific physical predictions can change significantly.