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A full mathematical framework for the analysis and production of localized magnetic field profiles is presented. Of primary use in the production of highly homogeneous fields for nuclear magnetic resonance studies, the paper details the analysis of fields in terms of spherical harmonics, describes how field plotting in the appropriate manner may be used to obtain a direct measure of which harmonics are present, and shows how to combine basic "building blocks" to produce the various lower-order zonal and tesseral harmonics. "Building blocks" described include coils, arcs, and sinusoids of current as well as rings and arcs of steel. The use of shaped magnets is also briefly mentioned. Attention is drawn to the presence, in high-order designs, of possibly dominant lower orders of harmonics created by errors in fabrication. The goal of the paper is to present a design philosophy, backed by the appropriate mathematics, which is applicable to the variety of situations encountered in magnet design. Practical examples of correcting coils and "shims" are also given.

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