I'll give a brief introduction to K-homology theory, the generalized homology theory that is dual to Atiyah-Hirzebruch K-theory, as it is viewed from C*-algebra theory and from index theory. As I hope to illustrate, the two perspectives are in many respects complementary. Then I'll turn to the [Q,R]=0 problem. It is easy to sketch a simple connection between the C*-algebraic version of K-homology and the analytic proof of [Q,R]=0. But the geometric side of the story remains quite mysterious.