  Our purpose is to introduce in the setting of locally compact groupoids the
analogues of the well known equivalent definitions of exactness for discrete
groups. We study the relations between these notions. The best results are
obtained for a class of \'etale groupoids that we call inner amenable, since
for locally compact groups this notion coincides with the notion of inner
amenability. We give examples of such groupoids which include transformation
groupoids associated to actions of discrete groups by homeomorphisms on locally
compact spaces. We have no example of \'etale groupoids which are not inner
amenable. For inner amenable \'etale groupoids we extend what is known for
discrete groups in proving the equivalence of six natural notions of exactness:
(1) strong amenability at infinity; (2) amenability at infinity; (3) nuclearity
of the uniform (Roe) algebra of the groupoid; (4) exactness of this
$C^*$-algebra; (5) exactness of the reduced $C^*$-algebra; (6) exactness of the
groupoid in the sense of Kirchberg-Wassermann.
  We give several illustrations of our results and review the results obtained
by several authors that highlight the crucial role of exactness in order to
clarify the relation between the amenability of a groupoid and the fact that
its full and reduced $C^*$-algebras coincide. We end our paper with open
questions and an appendix on fibrewise compactifications because our work
requires to extend from discrete groups to \'etale groupoids the notion of
Stone-Cech compactification on which the groupoid acts.
