It is well known that an elliptic operator on a compact manifold is aFredholm operator, and the celebrated Atiyah-Singer index theoremcomputes the Fredholm index by using certain topological data. In contrast,an elliptic operator on a non-compact manifold is in generalno longer Fredholm in the usual sense, but "Fredholm" in a generalizedsense. One can define its generalized Fredholm index, called higher index,by using operator K-theory. It turns out that, on a general non-compactcomplete Riemannian manifold, the higher index of an elliptic operator isclosely related with the large scale geometry, also called coarse geometry,of the manifold. The coarse Baum-Connes conjecture is a program tocompute the higher indices of elliptic operators on non-compact spaces byusing certain topological invariants much in the spirit of Atiyah-Singer. In this talk, we will first introduce the basic ideas of coarse geometry ofmetric spaces and the associated operator algebras encoding the coarsegeometry.Then we will give a survey on some of recent development of higher indextheory, its applications, and its fascinating connection to the geometry ofgroups,expander graphs and various other metric spaces.