The nucleus of a galaxy orbits around the mass centroid. Orbital motions appear overstable in numerical experiments started with a galaxy's nucleus at rest atop its mass centroid. The amplitude doubles in 6-10 orbital periods. Orbits precess, nutate, and change their amplitudes, but they keep fairly constant periods. Orbital periods are in resonance with local particle motions, and amplitudes reach a core radius. This resonance suggests that center motions are a local, rather than a global, phenomenon. The overstability implies that a galaxy cannot be formed in nature with its nucleus at rest atop its mass centroid, and that nuclei orbit the mass centroid in real galaxies. These center motions should show up observationally as a shift of the nucleus away from the center defined by nearby isophotes. Off-center nuclei have been reported in many galaxies (e.g., M33, M101, NGC 3379, NGC 3384). Other kinds of observations confirmed the picture of nonsteady galactic centers as well. Gas trapped in moving nuclear regions of a galaxy should show strange flow patterns with possible shocks. The nuclear regions of galaxies including Milky Way and of globular clusters are not likely to be in a static steady state.