Black holes are one of the fundamental predictions in the theory of General Relativity, and the ones of most interest are the Kerr spacetimes, the metrics of which describe a family of rotating, stationary, axisymmetric, asymptotically flat solutions to the vacuum Einstein equations. One of the central problems in Mathematical General Relativity is the Kerr Stability Conjecture, which aims to show that the family of Kerr spacetimes is nonlinearly stable against small perturbations. I will discuss my own work on this conjecture, along with the recent advances made toward it.