ABSTRACT Since the seminal work by Chandrasekhar, the random walk scheme has been extensively developed both by expansion of new theoretical approached and through applications to a vast deversity of experimental situations. In particular the original motivation for describing the transport by hopping conduction by Scher and Lax found a natural basement on the Montroll-Weiss continous time random walk on a lattice. We present a comprehensive review based on the master equation with transition rates characterized by a given probability distribution. Thus, models of disorder are described by the properties of the distribution and specific symmetries in the underlying lattice. The aim of the theory is to find an approximation to calculate the mean value of the Green function of the problem. A unified description is presented, based on a diagrammatic technique, in order to compare the different approximations to tackle the problem of transport in disordered media. Important tools to solve this nonequilibrium problem are the effective medium approximation and non-Markovian random walk schemes. Particular emphasis is put on the calculation of the frequency dependence behavior of the electric conductivity, first passage times and residence times. The influence of boundary conditions on the Green function is put in correspondence with other interesting subjects like the first passage time problem and related ones. Examples reviewed include studies of diffusion in pure and drifted lattices in presence of different kinds of disorder and anisotropies and of the lattice.
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