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 timedependent 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.