Testing quantum theory with thought experiments

Quantum mechanics is one of our most successful physical theories; its predictions agree with experimental observations to an extremely high accuracy. However, the bare formalism of quantum theory does not provide straightforward answers to seemingly simple questions: for example, how should one model systems that include agents who are themselves using quantum theory? These foundational questions may be investigated with a theorist's tool -- the thought experiment. Its purpose is to turn debates about the interpretation of quantum theory into actual physics questions. In this article, we give a state-of-the-art overview on quantum thought experiments involving observers, from the basic Wigner's friend to the recent Frauchiger-Renner setup, clarifying their interpretational significance and responding to objections and criticism on the way.