Nonlinear fields are so important in modern accelerators or storage rings that they have to be included in the basic lattice design for both chromaticity correction and aperture determination. The power of prediction of a nonlinear theory is not yet firm enough and numerical simulations are necessary. These lectures deal with the principles of particle tracking, the equations of the betatron motion and the envelope distortions and tune shifts induced by multipole fields. The theory is applied to the correction of beam envelope distortions produced by chromaticity sextupoles in a large acceptance machine, the Antiproton Collector.