The Petrov classification categorizes vacuum spacetimes by the algebraic structure of the Weyl tensor — the "free" part of the Riemann tensor encoding gravitational radiation. Using principal null directions (PNDs), spacetimes fall into types I (4 distinct PNDs), II (1 double + 2 single), D (2 double), III (1 triple + 1 single), N (1 quadruple), and O (flat). The Schwarzschild and Kerr black holes are type D; gravitational waves far from sources are type N. The Goldberg-Sachs theorem: a vacuum spacetime is algebraically special (type II, D, III, N) if and only if it admits a shear-free null geodesic congruence.