Wave propagation and radiation processes are common phenomena across various physical theories. In this context, I will present the mathematical and physical aspects of the so-called hyperboloidal framework, a concept developed upon Penrose’s seminal work on the “Conformal Treatment of Infinity”. Over the past decades, the hyperboloidal approach has emerged as a crucial tool in black hole perturbation theory, but its fundamental concepts should apply to several scenarios modelled by wave-like behaviour. The first part of this talk provides an overview of the foundational principles of the hyperboloidal framework, emphasizing its role in accurately describing wave behaviour near infinity. In the second part, I will discuss some contributions of the formalism to gravitational-wave physics.