We have analyzed field line resonances of Alfv¿n waves in a rectangular box model with a straight uniform magnetic field but three-dimensionally varying density. Field line resonances are shown to exist even with this three-dimensional nonuniformity. For a given wave frequency we can construct the surface on which the resonance occurs and derive the local form of the singular solution. Magnetic perturbations are found to lie predominantly in the resonant surface. In the presence of azimuthal inhomogeneities the present theory could explain why some satellite measurements show geomagnetic pulsations of comparable magnitude in radial and azimuthal components. ¿ American Geophysical Union 1992