Lie groupoids and index theory - a brief outline

In this introductory talk we will discuss Lie groupoids and give examples of how they relate to several objects of classical differential geometry such as principal bundles, Lie group actions and foliations.
We will also see some of the ways in which groupoids can appear in the study of singular (i.e. not smooth) spaces such as orbit spaces of Lie group actions, leaf spaces of foliations, and pseudomanifolds with conical singularities.
Finally, we will see a brief outline of how groupoids can be useful when proving index theorems.
A reference for this talk is Debord-Lescure, Index theory and groupoids - arXiv:0801.3617