One of the first steps in studying three-dimensional manifolds is the Poincaré conjecture. This conjecture states that every simply connected, closed, and orientable three-dimensional manifold is homeomorphic to the 3-sphere. In the 1980s, Thurston formulated a stronger conjecture than the Poincaré conjecture. Thurston's Geometrization Conjecture states that every closed, orientable, and prime three-dimensional manifold can be cut along a number of tori such that the interior of each resulting manifold has one of Thurston's eight geometric structures with finite volume. In dimension 2, the Geometrization Conjecture reduces to the uniformization theorem. Our goal is to study the decomposition theorems for three-dimensional manifolds (the Prime Decomposition and JSJ Decomposition) and then examine Thurston's eight geometries to precisely state the Geometrization Conjecture. Following this, we will explore some consequences of this conjecture, such as the fact that the fundamental group is almost a complete invariant for closed three-dimensional manifolds.

Some references:
Three-Dimensional Geometry and Topology, William P. Thurston and Silvio Levy
An Introduction to Geometric Topology, Bruni Martelli