We review our state of knowledge of coronal element abundance anomalies in the Sun and stars. We concentrate on the first ionization potential (FIP) effect observed in the solar corona and slow-speed wind, and in the coronae of solar-like dwarf stars, and the "inverse FIP" effect seen in the corona of stars of later spectral type; specifically M dwarfs. These effects relate to the enhancement or depletion, respectively, in coronal abundance with respect to photospheric values of elements with FIP below about 10~eV. They are interpreted in terms of the ponderomotive force due to the propagation and/or reflection of magnetohydrodynamic waves in the chromosphere. This acts on chromospheric ions, but not neutrals, and so can lead to ion-neutral fractionation. A detailed description of the model applied to closed magnetic loops, and to open field regions is given, accounting for the observed difference in solar FIP fractionation between the slow and fast wind. It is shown that such a model can also account for the observed depletion of helium in the solar wind. The helium depletion is sensitive to the chromospheric altitude where ion-neutral separation occurs, and the behavior of the helium abundance in the closed magnetic loop strongly suggests that the waves have a coronal origin. This, and other similar inferences may be expected to have a strong bearing on theories of solar coronal heating. Chromospheric waves originating from below as acoustic waves mode convert, mainly to fast mode waves, can also give rise to ion-neutral separation. Depending on the geometry of the magnetic field, this can result in FIP or Inverse FIP effects. We argue that such configurations are more likely to occur in later-type stars (known to have stronger field in any case), and that this explains the occurrence of the Inverse FIP effect in M dwarfs.

Review paper submitted to Living Reviews in Solar Physics. 74 pages. Some material revised and updated from astro-ph/0405230, arXiv:0901.3350, arXiv:1110.4357