This is an introductory talk on K-homology and its role in index theory. K-homology is the homology theory "dual" to K-theory. The models of K-homology that are relevant for index theory are analytic K-homology and geometric K-homology. Analytic K-homology is based on ideas of Atiyah, and fully developed by Kasparov. Cycles in analytic K-homology are functional analytic objects, generalizing elliptic operators. Geometric K-homology is a development of bordism theory, and is due to Baum and Douglas. For finite CW complexes geometric and analytic K-homology are isomorphic. K-homology provides a powerful framework for index theory. In this talk I will introduce K-homology, and in this context consider the Atiyah-Singer index formulas, the index formula for Toeplitz operators due to Boutet de Monvel, and a recent index formula for operators in the Heisenberg calculus for contact manifolds (due to Baum and myself).