Global hyperbolicity is a classical and well-known concept, whichlies in the core of General Relativity. Here we discuss brieflyfive approaches to this concept. They yield different definitionswhich become natural in diverse contexts: the initial valueproblem, singularity theorems, existence of maximizing causalgeodesics, possibility to split globally the spacetime, causalboundaries. The neat formulation and definitive equivalencebetween these definitions have been completed only recently. Avery brief summary is presented.