Étale groupoids and their associated $C^*$-algebras, following Renault’s pioneering work, provide a powerful and unifying framework for modeling operator algebras. In this context, a natural and highly expressive way to generate groupoids is through partial actions of discrete groups. This motivated a fundamental structural question, often referred to as Exel's problem: Is every locally compact Hausdorff étale groupoid isomorphic to a transformation groupoid associated with a partial action?

In this talk, we will explore the successes, the boundaries, and the ultimate limits of groupoid modeling for $C^*$-algebras. We will begin with a positive answer to Exel's question in the realm of highly structured spaces, showing that the path groupoids of many higher-rank graphs — as well as ample models for UCT Kirchberg algebras — can be perfectly realized via partial actions. However, the dynamic and topological rigidity of partial actions eventually imposes severe constraints. We will present the first explicit examples of locally compact Hausdorff étale groupoids that do not arise from partial actions. These obstructions stem from algebraic properties, such as a lack of inner amenability in groupoids built from non-amenable residually finite groups, as well as topological barriers found in Deaconu-Renault groupoids with connected unit space.

Finally, relaxing the requirement of partial actions and allowing for general twisted étale groupoids, we investigate the absolute limits of Renault's modeling paradigm. We will establish that the algebra of all bounded linear operators on an infinite-dimensional Hilbert space, $B(H)$, completely resists this framework and cannot be realized as a twisted groupoid $C^*$-algebra.

This talk is based on joint works with Julian Kranz (Münster), and with Luiz Felipe Garcia (Florianópolis) and Tomás Pacheco (Lisbon).