An essential ingredient of AdS/CFT, dS/CFT and other dualities is a geometric notion of scattering that refers to asymptotics rather than, say, infinite time limits. Though one expects non-perturbative versions to exist in the case of linear quantum fields (and non-linear classical fields), this has been rigorously implemented in Lorentzian settings only relatively recently. The goal of this talk will be to give an overview in different geometrical setups, including asymptotically Minkowski, de Sitter and Anti-de Sitter spacetimes. In particular I will discuss recent results on classical scattering and particle interpretations, compare them with the setup of conformal scattering and explain how they can be used to construct "in-out" Feynman propagators (based on joint works with Christian Gérard and András Vasy).
