A decomposition of projective manifolds X by means of canonically bimeromorphically defined fibrations will be presented (appeared in Ann. Inst. Fourier, 53 (2004), 499-665). The fibres of these fibrations are orbifolds either of general type, or with kappa = 0, or with kappa_+ = (minus infinity), the latter being conjecturally an orbifold version of rational connectedness.
This decomposition is done in two steps: the first (and most fundamental) is a fibration c_X: X --> C(X), the "core" of X, with fibres "special," and orbifold base of general type. The second step canonically expresses the core (and so any special manifold) as a tower of fibrations having orbifold fibres with either kappa = 0, or kappa_+ = (minus infinity).
The core should also split X at the arithmetic and Kobayashi pseudometric levels. This will be discussed.
The new (quite simple) ingredient is the notion of orbifold base of a fibration, constructed out its multiple fibres.