Using the Green function of a random walk on a discrete group, we may define a distance which happens to have some nice geometric properties and is also useful to describe, in some cases, asymptotic behavior of the random walk itself. We will define this Green distance, explain some finite range properties (volume of the associated balls, comparison with the usual graph distance) as well as some asymptotic properties (relation between the asymptotic entropy and the associated rate of escape, dimension of the harmonic measure in the case of hyperbolic groups).