Thurston's geometrisation conjecture

Let $M$ be an irreducible orientable compact 3-manifold with possibly empty boundary consisting of tori. A geometric 3-manifold is one whose interior has a finite-volume complete geometric structure modelled on one of

$$S^3,\ \mathbb{R}^3,\ \mathbb{H}^3,\ S^2\times\mathbb{R},\ \mathbb{H}^2\times\mathbb{R},\ \operatorname{Nil},\ \operatorname{Sol},\ \widetilde{\operatorname{SL}_2}.$$

The geometrisation conjecture. Every block of the geometric decomposition of $M$ is geometric. Proposed by Thurston in 1982, this conjecture was proved by Perelman in 2002, providing the geometric classification underlying the modern understanding of compact 3-manifolds.