In describing properties of disordered media, physicists have long been interested in the behaviour of random walks on random graphs that arise in statistical mechanics, such as percolation clusters and various models of random trees. Random walks on random graphs are also of interest to computer scientists in studies of complex networks. In 'critical' regimes, many of the canonical models exhibit large-scale fractal behaviour, which mean it is often a challenge to describe their geometric properties, let alone the associated random walks. However, in
recent years, the deep connections between electrical networks and stochastic processes have been advanced so that tackling some of the key examples of random walks on random graphs is now within reach. In this talk, I will introduce some recent work in this direction, and describe some prospects for future developments.