The principal effects of planetary perturbations on the orbit of a sungrazing comet are derived, both from a simplified general theory and by direct numerical integration of typical trajectories. New orbit determinations are made for the three most recently observed members of the Kreutz group: comets 1965 VIII, 1963 V, and 1945 VII. For comet 1965 VIII the results are obtained in terms of both general relativity theory and the Brans-Dicke theory, but the observations are insufficient to enable one to decide which theory is correct. Some remarks are made about the less certain members of the comet group, and evidence is provided for dividing the comets into two distinct subgroups. Studies on the past motions of comets 1882 II and 1965 VIII reveal it as virtually certain that these comets separated from each other at their previous perihelion passage, and it seems possible that they were then observed as the comet of 1106. Not all the members of each subgroup can be explained so simply, but it is probable that the subsidiary separations took place not more than two revolutions ago. The two subgroups can presumably be explained by a similar split some 10 to 20 revolutions ago, although a few other possible explanations are touched upon in the final section. An ephemeris is provided for use in searching for further members of the group.