In QFT on curved spacetimes, a central difficulty is how to reconcile local properties of states (short-distance behaviour of N-point functions) with global ones (positivity, spacetime or gauge symmetries, asymptotic convergence to vacuum or thermal state, boundary conditions, etc.). In the last few years, these difficulties have been overcome in a wide range of situations, leading in particular to rigorous descriptions of thermal effects on black hole spacetimes. The aim of the talk will be to present the new state-of-the-art and discuss the main conjectures, with a particular focus on: - distinguished states on black hole spacetimes and their thermal properties - Hadamard states and (generalized) Feynman propagators in scattering situations - the Reeh-Schlieder property - states with good holographic properties on AdS spacetimes (Joint work with Ch. Gerard)