  We shall show how to decompose, by functorial and canonical fibrations,
arbitrary $n$-dimensional complex projective {Although the geometric results
apply to compact K\" ahler manifolds without change, we consider here for
simplicity this special case only.} varieties $X$ into varieties (or rather `
geometric orbifolds\rq $)$ of one of the three \pure geometries determined by
the `sign' (negative, zero, or positive) of the canonical bundle. These
decompositions being birationally invariant, birational versions of these \pure
geometries, based on the \canonical (or ` Kodaira\rq $)$ dimension will be
considered, rather.
  A crucial feature of these decompositions is indeed that, in order to deal
with multiple fibres of fibrations, they need to take place in the larger
category of `geometric orbifolds' $(X| \Delta)$. These are `virtual ramified
covers' of varieties, which `virtually eliminate' multiple fibres of
fibrations. Although formally the same as the `pairs' of the LMMP (see
\cite{kmm}, \cite{KM}, \cite{BCHM} and the references there), they are here
fully geometric objects equipped with the usual geometric invariants of
varieties, such as sheaves of (symmetric) differential forms, fundamental
group, Kobayashi pseudometric, integral points, morphisms and rational maps.
  It is intended to expose (with some addtional topics or developments), as
briefly and simply as possible, and essentially skipping the proofs, the main
content of arXiv:math/0110051 and arXiv:0705.0737 respectively published in
ANN. Inst. Fourier (2004), and to appear in J.Inst. Math. Jussieu.
