The Klein-Gordon or wave equation has rather unusual features if one imposes so-called Feynman asymptotic conditions. It enjoys a theory that in some ways resembles elliptic settings (think Fredholm operators, limiting absorption principle, standing waves, etc.). In this talk I will give an overview of this theory, and explain how it arises in computing quantum corrections to General Relativity and in Lorentzian geometry. I will then present new results on asymptotically Minkowski spaces for Klein-Gordon equations with complex mass term. The talk is among others based on joint works with Nguyen Viet Dang (Sorbonne Université), Christian Gérard (Paris-Saclay) and András Vasy (Stanford).