  A global analysis of the experimental data on azimuthal asymmetries in
semi-inclusive deep inelastic scattering (SIDIS), from the HERMES and COMPASS
Collaborations, and in e+e- --> h1 h2 X processes, from the BELLE
Collaboration, is performed. It results in the extraction of the Collins
fragmentation function and, for the first time, of the transversity
distribution function for u and d quarks. These turn out to have opposite signs
and to be sizably smaller than their positivity bounds. Predictions for the
azimuthal asymmetry A_{UT}^{sin(phi_h + phi_S)}, as will soon be measured at
JLab and COMPASS operating on a transversely polarized proton target, are then
presented.
