For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show
the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of an
Iitaka fibration of W such that the pluricanonical system |mK_W| defines an Iitaka fibration. This is a consequence of a
more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs,
singularities, log canonical thresholds, adjunction, etc.