 The Bott periodicity theorem was a central theme in Atiyah's work on K-theory, and he repeatedly returned to it, reworking the proof several times to understand it better.With Bott he worked out an elementary proof, and gave another version of it in his book.With Bott and Shapiro he analysed the relation of Bott periodicity to the periodicity of Clifford algebras; although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood.