Published by the American Geophysical Union as part of the Geophysical Monograph Series, Volume 111.Using the concept of magnetic helicity, physicists and mathematicians describe the topology of magnetic fields: twisting, writhing, and linkage. Mathematically, helicity is related to linking integrals, which Gauss introduced in the 19th century to describe the paths of asteroids in the sky. In the late 1970s the concept proved to be critical to understand laboratory plasma experiments on magnetic reconnection, dynamos, and magnetic field relaxation. In the late 1980s it proved equally important in understanding turbulence in the solar wind and the interplanetary magnetic field. During the last five years interest in magnetic helicity has grown dramatically in solar physics, and it will continue to grow as observations of vector magnetic fields become increasingly sophisticated.