This volume contains contributions presented at an International Symposium on Conformal Groups and Conformal Structures held in August 1985 at the Arnold Sommerfeld Institute for Mathematical Physics in Clausthal. We hope that the wide range of subjects treated here will give a picture of the present status of the importance of the conformal groups, other related groups and associated mathematical structures (such as superconformal algebra, Kac-Moody algebras), and spin structures. Symmetry, with group theory and algebras as its mathematical model, has always played a crucial and significant role in the development of physical theories. One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the Poincare group. It opens the door to applications far beyond the standard kinematical framework provided by the local symmetries of flat space-time.