3-manifolds exhibit a phenomenon called a standard two-level decomposition.

- a connected sum decomposition, where every compact 3-manifold is the connected sum of a unique collection of prime three-manifolds
- the Jaco-Shalen-Johannson torus decomposition

- Separate a 3-manifold into its connected sum, and then each summand is reduced by its Jaco-Shalen-Johannson torus decomposition.

- Euclidean geometry
- Hyperbolic geometry
- Spherical geometry
- The geometry of S
^{2}x R - The geometry of H
^{2}x R - The geometry of
*SL*_{2}R - Nil geometry, or
- Sol geometry.

Every irreducible, compact 3-manifold falls into exactly one of the following categories:

- it has a spherical geometry
- it has a hyperbolic geometry
- The fundamental group contains a subgroup isomorphic to the free abelian group on two generators (this is the fundamental group of a torus).

Progress has been made in proving that 3-manifolds that should be hyperbolic are in fact so. Mainly this progress has been limited to checking examples and reduction to more seemingly tractable conjectures, e.g. Virtually Haken Conjecture.

The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow. In 1982, Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would smooth out any bumps in the metric, resulting in a metric of constant positive curvature, i.e. a spherical metric. He later developed a program to prove the Geometrization Conjecture by Ricci flow.

Grigori Perelman may have now solved the Geometrization Conjecture (and thus also the Poincaré Conjecture) but because this latter makes Perelman eligible for a million dollar Millennium Prize Problems his work will need to survive two years of systematic scrutiny before the conjecture(s) will be deemed to have been solved.