The Hanle effect concerns the modification of polarized resonance-line scattering by magnetic fields; thus, it can be used as a diagnostic of stellar magnetic fields. The Hanle effect has been used to determine the field strength and distribution of magnetic structures present in prominences of the Sun. To investigate its potential use in stellar astronomy, the simplified case of an optically thin axisymmetric ring illuminated by a stellar point source is considered. The results are then used to derive the polarization from polar plumes, equatorial disks, and spherical shells. The integrated line polarization is calculated for axisymmetric rings with a variety of magnetic field orientations, and in every case the polarization is proportional to sin2i (where i is the viewing inclination), just as in the zero field case. It is also found that the Hanle effect can significantly alter the integrated line polarization. In some cases the position angle of the polarization in the line can be rotated by 90 degrees relative to the zero field case. We consider the Hanle effect as a possible diagnostic of magnetic fields in stellar winds with prominent ultraviolet and visible resonance lines. For these lines the diagnostic has sensitivity in the range of 1-1000 G. The Zeeman effect is not normally applicable for diagnosing magnetic fields in stellar winds in the subkilogauss range; thus, the Hanle effect should provide an especially useful new method of determining magnetic fields in stars other than the Sun. Possibilities for measuring the fields in early-type stars using ultraviolet observations is discussed.