  We describe how Lie groupoids are used in singular analysis, index theory and
non-commutative geometry and give a brief overview of the theory. We also
expose groupoid proofs of the Atiyah-Singer index theorem and discuss the
Baum-Connes conjecture for Lie groupoids. With the help of the general
framework of Lie groupoids and related structures we survey recent progress on
problems which were outside the scope of the original work of Atiyah and
Singer. This includes the Atiyah-Singer type index problem for many classes of
non-compact manifolds (e.g. manifolds with a Lie structure at infinity). We
also consider generalizations of the pseudodifferential calculus on Lie
groupoids, e.g. for boundary value problems.
