Abstract: Topological K-theory was the first "extraordinary" cohomology theory introduced into algebraic topology. It is extraordinary not only in that it generalized ordinary cohomology theory but also because of its immediate success in solving important problems and in contributing to the development of new mathematical theory. Topological K-theory was developed in the late 1950's by Atiya and Hirzebruch, building on the work of Grothendieck. During the 1960's in particular it found a number of important applications by Atiyah and collaborators. The goal of my talk is to explain the basics of this classical theory and to outline some of its important applications, namely those concerning how it can be used to study Lie groups and their classifying spaces. The usefulness of equivariant K-theory is a particular theme of these applications.