  From the mid-1970s, Eberhard Kirchberg undertook a remarkable extensive study
of $C^*$-algebras exactness whose applications spread out to many branches of
analysis. In this review we focus on the case of groupoid $C^*$-algebras for
which the notion of exactness needs to be better understood. In particular some
versions of exactness play an important role in the study of the weak
containment problem, that is whether the coincidence of the full and reduced
groupoid $C^*$-algebras implies the amenability of the groupoid or not.
