Strong Cosmic Censorship (SCC) is a conjecture, envisioned by Penrose, over 50 years ago, concerning the deterministic character of General Relativity. In this talk, we'll start by showing how the study of the global structure of the Reissner-Nordström and Kerr black hole solutions motivates this conjecture. Then we'll examine some of the challenges one faces in obtaining a precise and robust formulation for SCC and briefly present the state of the art concerning the asymptotically flat case. To finish, we'll discuss how the inclusion of a positive cosmological constant leads us to the study of quasinormal modes in de Sitter black holes and how this reveals severe difficulties in upholding the SCC conjecture.

--