Modern hadron colliders are made with high‐field superconducting dipoles and quadrupoles that in general exhibit unintentional imperfections of the guiding field‐shape, due to construction tolerances, persistent currents, and saturation. Although the departure from linearity is typically small (a few units in 10−4at 1 cm from the axis), this phenomenon has a profound influence in the single‐particle beam dynamics. The non‐linear fields are at the origin of two effects: the betatron tunes change with the amplitude and the momentum of the circulating particles, and, for certain combinations of the horizontal, vertical, and synchrotron tunes, non‐linear resonances are excited. As a consequence, the stability of the particle motion can be considerably reduced, to the point that the dynamic aperture can become smaller than the physical aperture of the magnets. Analytical, numerical, and experimental approaches have been used to evaluate and possibly compensate mechanisms that lead to particle losses. © 1995American Institute of Physics