It is our aim to give a contemporary account of a small, but well-developed and useful, part of the immense mathematical field of (compact) transformation groups - namely that in which the main tools are ordinary cohomology theory and rational homotopy theory. Furthermore, except for occasional excursions, we shall restrict our attention to those groups for which these methods work best: these are tori and elementary abelian p-groups. (An elementary abelian p-group, or p-torus, where p is a prime number, is just a product of finitely many copies of the cyclic group of order p.) Torus and p-torus actions are of more than mere intrinsic interest, however: one can extrapolate to gain much useful information about more general group actions, often in a way similar to that in which the classification of compact connected Lie groups was achieved by studying the roots and weights associated with representations of maximal tori. Two important references where the reader can see such extrapolation at work are [Quillen, 1971a, b] and [Hsiang, 1975].

Our subject began with the work of P.A.Smith in the 1930s and 1940s; and consequently, it is often called P.A. Smith theory. Important developments were brought together in the Princeton seminar [Borel et al., 1960]. Later the subject received substantial clarification and inspiration, when, prompted by the work of Atiyah and Segal in equivariant K-theory, some ideas, which were implicit in the work of Borel, were reformulated in the succinct Localization Theorem proven independently by W.-Y. Hsiang and Quillen.