Rather than discussing some recent results on an analytic Groethendieck-Riemann-Roch theorem, I will focus on the development of K-homology by Baum and myself, which made this work possible. My talk will be both expository and reminiscences.

The topics covered will be the isomorphism between an analytic and geometric realization of K-homology along with an analytic realization for the even relative group. A final effort including Michael Taylor allows one to prove various index theorems, both that of Attiyah-Singer and Boutet de Monvil.