  We extend the symbol calculus and study the limit operator theory for
$\sigma$-compact, \'{e}tale and amenable groupoids, in the Hilbert space case.
This approach not only unifies various existing results which include the cases
of exact groups and discrete metric spaces with Property A, but also establish
new limit operator theories for group/groupoid actions and uniform Roe algebras
of groupoids. In the process, we extend a monumental result by Exel, Nistor and
Prudhon, showing that the invertibility of an element in the groupoid
$C^*$-algebra of a $\sigma$-compact amenable groupoid with a Haar system is
equivalent to the invertibility of its images under regular representations.
