The celebrated strong cosmic censorship conjecture in general relativity in particular predicts that the smooth Cauchy horizons present in the interior of explicit Reissner-Norstr”om and Kerr black hole solutions are unstable against perturbations. We present recent results regarding both stability (joint with M. Dafermos) and instability (joint with S.-J. Oh) of the smooth Cauchy horizon. The former result shows that for all small perturbations (without symmetry assumptions), the solution possesses a Cauchy horizon such that the metric remains continuous. In particular, if there are singularities in these perturbed spacetimes, they must be so-called weak null singularities.