Solenoid systems comprising three or four coils may be designed to produce magnetic fields of extremely high homogeneity, by eliminating the leading perturbation terms in the field series formula. Alternatively, by minimizing the field deviation over a specified region, a three-coil design may be produced to provide more economically a similar degree of homogeneity. Residual field irregularities may be eliminated by means of non-solenoidal windings which generate individually the leading terms in the spherical harmonic solution of Laplace's equation.