abstract: Globally hyperbolic spacetimes of constant curvature provide a nice parallel in the lorentzian context of the classical study of Kleinian groups in hyperbolic geometry. Nowadays, the classification of these spacetimes are well understood, at least in the regular case, ie. without particles. The goal of the minicourse is to provide the foundation of this theory, to present the link with Anosov representations of Gromov hyperbolic groups in SO(2,n) or SO(1,n) x R{1,n}, the question of the geometric convergence of level sets of time functions near the "initial singularity", and related open questions in the field.