Some main sequence A and B stars have strong, ordered magnetic fields. These Ap and Bp stars usually have anomalous chemical abundances, and often rather non-uniform distributions of at least some elements (e.g. He, Ca, Ti, Cr) over their surfaces. Maps of magnetic field structure may provide a means of testing theories of how large coherent fields form, and of probing large-scale hydrodynamic flows (meridional circulation) inside stars. Maps of distributions of various elements can help to elucidate the mechanisms, such as diffusion under competing influences of gravity and radiation, turbulence, meridional circulation, mass loss, and perhaps accretion from the interstellar medium, that lead to the distinctive abundances and surface abundance distributions. This kind of mapping is important as an aid to understanding how the Ap and Bp stars develop. It is even more important because the processes involved in producing Ap and Bp stars probably have significant effects on surface chemical abundances of “normal” upper main sequence stars, and so understanding the relevant physics is essential to correctly relating observed surface chemistry to stellar and galactic evolution. In this paper, efforts to map field and abundance structures in Ap stars are reviewed, and some of the principal results obtained thus far are discussed.