Abstract: In this paper we consider two models of random walks on a random rough surface. In the first model the growth and fluctuations of the surface are modelled by Edwards–Wilkinson linear model, in second case they are modelled by non-linear Kardar–Parisi–Zhang equation. Despite the nonlinearity of the second model, ordinary perturbation theory gives us trivial critical dimensions and the only way to get non-trivial results is due to non-perturbative effects.