The loop group (studied by Pressley and Segal) is the space of maps from the circle to a compact Lie group. It is an infinite-dimensional object which shares some properties with the Lie group. The based loop is the space of such maps which send the based point to the identity element. It shares some properties with coadjoint orbits of Lie groups (for example, flag manifolds). In particular the based loop group has a symplectic structure. In this lecture I will describe some results on the K-theory and equivariant K-theory of the based loop group of SU(2). I will also describe a Hamiltonian torus action.