This research tract contains an exposition of our research on bordism and differentiable periodic maps done in the period 1960-62. The research grew out of the conviction, not ours alone, that the subject of transformation groups is in need of a large infusion of the modern methods of algebraic topology. This conviction we owe at least in part to Armand Borel; in particular Borel has maintained the desirability of methods in transformation groups that use differentiability in a key fashion [9, Introduction], and that is what we try to supply here. We do not try to relate our work to Smith theory, the homological study of periodic maps due to such a large extent to P. A. Smith; for a modern development of that subject which expands it greatly see the Borel Seminar notes [9]. It appears to us that our work is independent of Smith theory, but in part inspired by it. We owe a particular debt to G. D. Mostow, who pointed out to us some time ago that it followed from Smith theory that an involution on a compact manifold, or a map of prime period [italic lowercase]p on a compact orientable manifold, could not have precisely one fixed point. It was this fact that led us to believe it worthwhile to apply cobordism to periodic maps.