A comprehensive review is presented of the mathematical methods used to represent magnetic fields in the Earth's magnetosphere, of the way existing data-based models use these methods and of the associated problems and concepts. The magnetic field has five main components: the internal field, the magnetopause, the ring current, the tail and Birkeland currents. Methods of representing separately each of these are discussed, as is the deformation of magnetic fields; Appendix B traces the connection between deformations and the Cauchy integral. A summary section lists the uses of data-based models and their likely future evolution, and Appendix A supplements the text with a set of problems.