  This is an expository paper which gives a proof of the Atiyah-Singer index
theorem for elliptic operators. Specifcally, we compute the geometric K-cycle
that corresponds to the analytic K-cycle determined by the operator. This paper
and its companion ("K-homology and index theory II: Dirac Operators") was
written to clear up basic points about index theory that are generally accepted
as valid, but for which no proof has been published. Some of these points are
needed for the solution of the Heisenberg-elliptic index problem in our paper
"K-homology and index theory on contact manifolds" Acta. Math. 2014.
