We discuss recent results on the essential self-adjointness of Klein-Gordon type operators on several classes of spacetimes. The first one is the asymptotically flat spacetime, which was studied previously by A. Vasy (J. Spectral Theory 2020) and by us (Ann. H. Lebesgue 2021), but we present a new simpler proof (Ann. H. Poincaré
2023). We also discuss the essential self-adjointness for the asymptotically static spacetime, which is Cauchy compact (Comm. Math. Phys. 2023). These results are joint work with Kouichi Taira (Ritsumeikan University).