Index theory and Groupoids

This paper collects the notes of a serie of lectures given by the two authors during the summer school "Geometric and topological methods for Quantum Field Theory" at Villa de Leyva, Colombia, summer 2007. These lecture notes are mainly devoted to a proof using groupoids and $KK$-theory of Atiyah-Singer index theorem on compact smooth manifolds. We will present an elementary introduction to groupoids, $C^*$-algebras, $KK$-theory and pseudodifferential calculus on groupoids. We will finish by showing that the point of view adopted here generalizes to the case of conical pseudo-manifolds.