  So far, the most magnificent breakthrough in mathematics in the 21st century
is the Geometrization Theorem, a bold conjecture by William Thurston
(generalizing Poincar\'e's Conjecture) and proved by Grigory Perelman, based on
the program suggested by Richard Hamilton. In this survey article, we will
explain the statement of this result, also presenting some examples of how it
can be used to obtain interesting results in differential geometry.
