  We present some of the recent results and open questions on the causality
problem in General Relativity. The concept of singularity is intimately
connected with future trapped surface and inner event horizon formation. We
offer a brief overview of the Hawking Penrose singularity theorems and discuss
a few open problems concerning the future Cauchy development (domain of
dependence), breakdown criteria and energy conditions for the horizon
stability. A key question is whether causality violating regions, generating a
Cauchy horizon are allowed.
  We raise several questions concerning the invisibility and stability of
closed trapped surfaces from future null infinity and derive the imprisonment
conditions. We provide an new perspective of the causal boundaries and
spacelike conformal boundary extensions for time oriented Lorentzian manifolds
and more exotic settings.
