  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.
