The importance of the singularity theorems in Lorentzian Geometry, which give
su cient conditions on a spacetime entailing the incompleteness of its null or
timelike geodesics, is well known. But equally important is to ascertain the
rigidity of their conclusions if certain speci c key assumptions in these theorems
are removed or weakened, a problem related to the genericity of the singularities
in physically motivated contexts. In this talk I shall review the general setting
of rigid singularity theorems and will present some examples, including some
quite recent ones.