The hyperboloidal framework builds upon Penrose's seminal work on the "Conformal Treatment of Infinity." It's an elegant concept that enables us to approach the black hole event horizon and the wave zone "at the same time". Over the decades, it has become an indispensable tool in black hole perturbation theory, particularly when considering an expansion beyond the linear regime. In this presentation, I will delve into the latest advancements in the field. This includes introducing the so-called Minimal Gauge, a simple strategy to construct hyperboloidal slices and exploring its applications in gravitational wave physics. After reviewing contributions to fundamental concepts in the ringdown phase, I'll discuss technical advances in the modeling of extreme mass ratio inspirals through the self-force program.
