High-order radiofrequency transmitter coils are described that permit, in the context of NMR imaging, a knowledgeable trade-off between coil complexity and the degree of inhomogeneity over a specified volume of space. A mathematical formulation of high-order coils is developed for an axially symmetric polarizing dc field and transverse rf field. The field is created by a number of longitudinal current filaments with equal angular separations on the wall of a cylinder. In its most general form, each current can be independently specified. Circularly or linearly polarized fields can be created. Quadrature and saddleshaped coils emerge as special cases of the general formulation. The development has been tested experimentally and good agreement found. High-order coils can also be used as receivers, permitting more equal weighting of the sensitivities of each pixel.