Phenomena occurring in a particular class of nonlinear oscillatory systems—zero-dispersion systems—are reviewed for cases with and without damping while the system is driven either by random fluctuations (noise), or by a periodic force, or by both together. Zero-dispersion (ZD) systems are those whose frequency of oscillation ω possesses an extremum as a function of energy E. Oscillations at energies close to the extremal energy Em, where the “frequency dispersion” dω/dE is equal to zero, correlate with each other for very long times, to some extent like in a harmonic oscillator. But unlike the latter, the correlation time decreases as the energy shifts away from Em. It is the combination of this local harmonicity, with the fact that a perturbation can cause transitions between strongly and weakly correlated behaviour, that gives rise to the rich manifold of interesting ZD phenomena that are reviewed. A diverse range of physical systems may be expected to exhibit ZD behaviour under particular circumstances. Examples considered in detail include superconducting quantum interference devices, the 2D electron gas in a magnetic superlattice, axial molecules, electrical circuits, particle accelerators, impurities in lattices, relativistic oscillators, and the Harper oscillator. The ZD effects to be anticipated in quantum systems are also discussed. Each section ends with a suggested outlook for future research.