Explore the fascinating world of singularities in general relativity through this comprehensive lecture by Jonathan Luk. Delve into Lorentzian geometry and Einstein equations, examining the fundamental theorem of general relativity and key concepts like Minkowski and Schwarzschild spacetimes. Investigate Penrose's theorem, asymptotically flat initial data, and the Kerr spacetime, while considering the strong cosmic censorship conjecture and stability of Kerr exterior. Analyze blue shift instability, spacelike singularity conjecture, and the Reissner-Nordström spacetime. Discover solutions arising from spherically symmetric data, weak null singularities, and the formation of trapped surfaces in vacuum. Conclude by examining black hole spacetimes arising from gravitational collapse and the potential breakdown of weak null singularities.