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We get asymptotics for the volume of large balls in an arbitrary locallycompact group G with polynomial growth. This is done via a study of thegeometry of G and a generalization of P. Pansu's thesis. In particular, we showthat any such G is weakly commensurable to some simply connected solvable Liegroup S, the Lie shadow of G. We also show that large balls in G have anasymptotic shape, i.e. after a suitable renormalization, they converge to alimiting compact set which can be interpreted geometrically. We then discussthe speed of convergence, treat some examples and give an application toergodic theory. We also answer a question of Burago about left invariantmetrics and recover some results of Stoll on the irrationality of growth seriesof nilpotent groups.

Geometry of locally compact groups of polynomial growth and shape of large balls