The dynamics of magnetic moments in nanostructures is closely linked to that of gyroscopes. The Landau-Lifshitz-Gilbert equation describes precession and relaxation but does not include nutation. Both precession and relaxation have been observed in experiments, in contrast to nutation. The extension of the atomistic Landau-Lifshitz-Gilbert equation by a nutation term allows us to study the significance of nutation in magnetization dynamics of nanostructures: for a single magnetic moment, a chain of Fe atoms, and Co islands on Cu(111). We find that nutation is significant at low-coordination sites and on the time scale of about 100 fs; its observation challenges strongly today's experimental techniques.