  We study the transverse momentum dependent (TMD) evolution of the Collins
azimuthal asymmetries in $e^+e^-$ annihilations and semi-inclusive hadron
production in deep inelastic scattering (SIDIS) processes. All the relevant
coefficients are calculated up to the next-to-leading logarithmic (NLL) order
accuracy. By applying the TMD evolution at the approximate NLL order in the
Collins-Soper-Sterman (CSS) formalism, we extract transversity distributions
for $u$ and $d$ quarks and Collins fragmentation functions from current
experimental data by a global analysis of the Collins asymmetries in
back-to-back di-hadron productions in $e^+e^-$ annihilations measured by BELLE
and BABAR Collaborations and SIDIS data from HERMES, COMPASS, and JLab HALL A
experiments. The impact of the evolution effects and the relevant theoretical
uncertainties are discussed. We further discuss the TMD interpretation for our
results, and illustrate the unpolarized quark distribution, transversity
distribution, unpolarized quark fragmentation and Collins fragmentation
functions depending on the transverse momentum and the hard momentum scale. We
make detailed predictions for future experiments and discuss their impact.
