I will introduce Lie groupoids and some of their associated operator algebras. Then I will discuss how these operator algebras can be used to model and study various operators appearing in geometric applications and in other applications. In particular, I will show how the concept of 'exhausting families' of representations, can be used to obtain characterizations of the Fredholm operators using groupoids. This is conveniently done using the so called `Fredholm groupoids', whose study turns out to be closely related to the (generalized) Effros-Hahn conjecture for groupoids. The results of this talk are based also on a joint work with Nicolas Prudhon.