AbstractHemispherical gradient coils offer an open geometry that is well suited to imaging of the human brain. The windings of a hemispherical gradient coil are on average closer to the target region than those of a comparable cylindrical coil, and consequently hemispherical coils can produce higher efficiency at fixed inductance. The mathematical formalism needed for the design of hemispherical gradient coils is described here, including expressions relating the current distribution on the hemisphere to the magnetic field generated, as well as the stored energy and power dissipation. In addition, expressions for the torque experienced by the current distribution in the presence of the main magnetic field have been derived and used to develop an approach allowing the design of torque‐balanced, hemispherical transverse gradient coils. Hemispherical coil designs suitable for brain imaging are presented and shown to have improved performance compared with their cylindrical counterparts. Small, prototype, hemispherical z‐ and x‐gradient coils have been constructed and tested in phase‐mapping experiments at 3 T. The experimental results show good agreement with theoretical predictions, validating the mathematical expressions used in the coil design process. The formalism also allows the design of coils wound on a complete spherical surface, and the performance of such coils is additionally described here. Magn Reson Med, 2005. © 2005 Wiley‐Liss, Inc.