It follows from work of Perelman that any finite-time singularity of the Ricci flow on a compact 3-manifold is modeled on an ancient ϰ-solution.
We prove that every non-compact ancient ϰ-solution in dimension 3 is isometric to a family of shrinking cylinders (or a quotient thereof), or to the Bryant soliton. This confirms a conjecture of Perelman.