We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansus thesis. In particular, we show that any such G is weakly
commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups.
