This chapter is an up-to-date account of results on globally hyperbolic spacetimes, and serves several purposes. We begin with the exposition of results from a foundational level, where the main tools are order theory and general topology, we continue with results of a more geometric nature, and we conclude with results that are related to current research in theoretical physics. In each case, we list a number of open questions and formulate, for a class of spacetimes, an interesting connection between global hyperbolicity of a manifold and the geodesic completeness of its corresponding space-like surfaces. This connection is substantial for the proof of essential self-adjointness of a class of pseudo differential operators, that stem from relativistic quantum field theory.
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