In this paper we study the average distance in weighted graphs. More precisely, we consider assignments of families of non-negative weights to the edges. The aim is to maximise (minimise, respectively) the average distance in the resulting weighted graph. Two variants of the problem are considered depending on whether the collection of weights is fixed or not. The main results of this paper are the NP-completeness of the maximising version of the problem when the collection of weights is fixed, and an upper bound on the mean distance for weighted multigraphs with prescribed edge-connectivity. 