The motions of elliptical or polygonal vortex rings have been numerically simulated by using 3-dimensional vortex method. It is found that the distance that an elliptical vortex ring travels during its one periodical motion is the same only if the initial area of the ring is the same irrespective of its geometrical shape, i.e., its flatness. Also the motion of a pair of elliptical vortex rings is simulated, and its successive behaviors are graphically presented. Finally, motions of polygonal vortex rings are treated to examine the effect of the corner of the polygon.