  We construct examples of two-step and three-step nilpotent Lie groups whose
automorphism groups are `small' in the sense of either not having a dense orbit
for the action on the Lie group, or being nilpotent (the latter being
stronger). From the results we get also new examples of compact manifolds
covered by two-step simply connected Lie groups, which do not admit Anosov
automorphisms.
