Topological K–theory, the first generalized cohomology theory to be studied thoroughly, was introduced around 1960 by Atiyah and Hirzebruch, based on the Periodicity Theorem of Bott proved just a few years earlier. In some respects K–theory is more elementary than classical homology and cohomology, and it is also more powerful for certain purposes. Some of the best-known applications of algebraic topology in the twentieth century, such as the theorem of Bott and Milnor that there are no division algebras after the Cayley octonions, or Adams’ theorem determining the maximum number of linearly independent tangent vector fields on a sphere of arbitrary dimension, have relatively elementary proofs using K–theory, much simpler than the original proofs using ordinary homology and cohomology.