We have used Global Oscillation Network Group (GONG) magnetograms to characterize the changes in the photospheric longitudinal magnetic field during 15 X-class solar flares. An abrupt, significant, and persistent change in the magnetic field occurred in at least one location within the flaring active region during each event. We have identified a total of 42 sites where such field changes occurred. At 75% of these sites, the magnetic field change occurred in less than 10 minutes. The absolute values of the field changes ranged between 30 and almost 300 G, the median being 90 G. Decreases in the measured field component were twice as frequent as increases. The field changes ranged between 1.4 and 20 times the rms noise of the observations. In all but one equivocal case, the field changes occurred after the start of the flare. In all cases, the field changes were permanent. At least two-thirds of the field changes occurred in the penumbrae of sunspots. During three events for which simultaneous Transition Region and Coronal Explorer (TRACE) images are available, we have found excellent spatial and temporal correlation between the change in the magnetic field and an increase in brightness of the footpoints of flare ribbons, but not vice versa. Among many possible explanations for the observations, we favor one in which the magnetic field changes result from the penumbral field relaxing upward by reconnecting magnetic fields above the surface. One of the basic assumptions of flare theories is that the photospheric magnetic field does not change significantly during flares. These results suggest that this assumption needs to be re-examined.