Global hyperbolicity is a classical and well-known concept, which
lies in the core of General Relativity. Here we discuss briefly
five approaches to this concept. They yield different definitions
which become natural in diverse contexts: the initial value
problem, singularity theorems, existence of maximizing causal
geodesics, possibility to split globally the spacetime, causal
boundaries. The neat formulation and definitive equivalence
between these definitions have been completed only recently. A
very brief summary is presented.