Atiyah's \(KR\)-theory from his "K-Theory and Reality" paper (1966) establishes a unified view of complex and real $K$-theory and in particular gives a quite elegant proof of the periodicity theorem for \(KO\).
In this expository talk I will try to give a quick overview of the basic properties of this theory and afterwards focus on the proof of Bott periodicity in the real case via \(KR\)-theory.
Contrary to the approach of the original paper we will also encounter a proof that the Atiyah-Bott-Shapiro homomorphism is an isomorphism during this discussion.
The audience should be familiar with basic vector bundle theory, the basics of (complex) \(K\)-theory and maybe a few facts about Clifford algebras.