A random walk on a graph is a walk where at each vertex visited, an edge incident to the vertex is chosen at random, and the walk proceeds along the edge. 

A random walk on a graph represents a reversible Markov chain, whose transition probabilities depend on the degrees of the vertics. A graph is said to be recurrent on transient, according to wheather the corresponding Markov chain is recurrent or transient. We shall discuss how random walks on graphs can be used to classify Rievann surfaces, as to their hyperbolicity or parabolicity.