This talk will introduce the notion of equivariant $K$-theory with respect to groups that may not be compact. The main example in this talk will be the case of the loop group acting as the Gauge group on the space of connnections on the trivial $G$-bundle on $S^1$. The $K$-theory groups calculate representations of Loop groups, as proved in the recent work of Freed-Hopkins and Teleman. If time permits, I will explain how this example fits in a larger class of examples calculating representations of Kac-Moody groups.