  In its simplest form the Decomposition Theorem asserts that the rational
intersection cohomology of a complex projective variety occurs as a summand of
the cohomology of any resolution. This deep theorem has found important
applications in algebraic geometry, representation theory, number theory and
combinatorics. It was originally proved in 1981 by Beilinson, Bernstein,
Deligne and Gabber as a consequence of Deligne's proof of the Weil conjectures.
A different proof was given by Saito in 1988, as a consequence of his theory of
mixed Hodge modules. More recently, de Cataldo and Migliorini found a much more
elementary proof which uses only classical Hodge theory and the theory of
perverse sheaves. We present the theorem and outline the main ideas involved in
the new proof.
