  We address continuous-time quantum walks on graphs in the presence of time-
and space-dependent noise. Noise is modeled as generalized dynamical
percolation, i.e. classical time-dependent fluctuations affecting the tunneling
amplitudes of the walker. In order to illustrate the general features of the
model, we review recent results on two paradigmatic examples: the dynamics of
quantum walks on the line and the effects of noise on the performances of
quantum spatial search on the complete and the star graph. We also discuss
future perspectives, including extension to many-particle quantum walk, to
noise model for on-site energies and to the analysis of different noise
spectra. Finally, we address the use of quantum walks as a quantum probe to
characterize defects and perturbations occurring in complex, classical and
quantum, networks.
