  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.
