Our purpose is to introduce and study in the setting of locally compact groupoids the analogues of the well known equivalent definitions of exactness for discrete groups. The best results are obtained for a class of \'etale groupoids that we call weakly inner amenable, since for locally compact groups this notion is weaker than 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 weakly inner amenable. For weakly inner amenable \'etale groupoids we prove 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 groupoid in the sense of Kirchberg-Wassermann; (6) exactness of its reduced C^*-algebra. We end by several illustrations of our results and open questions.