This paper is a greatly expanded account of the preliminary material that appeared in Atiyah & Bott (1980). Part ofthe reason for the long delay between that paper and this present version was that new view-points emerged that added further interest to the topic and enabled us to provide proofs for what had earlier been conjectures. The length of this paper is due to our desire to present the several different aspects of the problem. We feel that this is justified because the main interest lies not so much in the actual applications as in the methods employed and the interaction between different approaches. Because ofits long gestation period and its consequent size, we feel that we owe the reader a substantial and leisurely introduction that puts the paper into historical perspective. In fact, on a personal level, one of the attractions of this subject to us is that it brings together algebraic vector bundles and Morse theory, topics to which we separately made contributions a quarter of a century ago (Atiyah 1955, 1957; Bott 1958, Bott &Samelson 1958). Even better, the catalyst that produced this interaction came from a quite unexpected quarter, namely that of theoretical physics in the form of the Yang-Mills equations.
 On Some Recent Development in Yang-Mills Theory.-  (with R. Seeley) Some Remarks on the Paper of Callias.-  Some Recollections from 30 Years Ago.-  Equivariant Morse Theory and the Yang-Mills Equation on Riemann Surfaces.-  Morse Theoretic Aspects of Yang-Mills Theory.-  An Equivariant Setting of the Morse Theory.-  Marston Morse and his Mathematical Works.-  Lectures on Morse Theory, Old and New.-  (with M.F. Atiyah) The Yang-Mills Equations Over Riemann Surfaces.-  (with M.F. Atiyah) The Moment Map and Equivariant Cohomology.-  On Topology and Other Things.-  On Some Recent Interactions Between Mathematics and Physics.-  On Characteristic Classes and Continuous Cohomology.-  Jerusalem Lectures.-  On the Fixed Point Formula and the Rigidity Theorems of Witten.-  On Mathematics and Physics.-  On Induced Representations.-  (with C. Taubes) On the Rigidity Theorem of Witten.-  Morse Theory Indomitable.-  Remarks Upon Receiving the Steele Career Prize at the Columbus Meeting of the A.M.S..- Errata.- Permissions — Volume 4.