  We study the question of local and global uniqueness of completions, based on
null geodesics, of Lorentzian manifolds. We show local uniqueness of such
boundary extensions. We give a necessary and sufficient condition for existence
of unique maximal completions. The condition is verified in several situations
of interest. This leads to existence and uniqueness of maximal spacelike
conformal boundaries, of maximal strongly causal boundaries, as well as
uniqueness of conformal boundary extensions for asymptotically simple
space-times. Examples of applications include the definition of mass, or the
classification of inequivalent extensions across a Cauchy horizon of the Taub
space-time.
