  By considering semi-inclusive deep-inelastic scattering and the
(complementary) $q_T$-spectrum for Drell-Yan lepton pair production we derive
the QCD evolution for all the leading-twist transverse momentum dependent
distribution and fragmentation functions. We argue that all of those functions
evolve with $Q^2$ following a \emph{single} evolution kernel. This kernel is
independent of the underlying kinematics and it is also spin-independent. Those
features hold, in impact parameter space, to all values of $b_T$. The evolution
kernel presented has all of its large logarithms resummed up next-to-next-to
leading logarithmic accuracy, which is the highest possible accuracy given
existing perturbative calculations. As a study case we apply this kernel to
investigate the evolution of Collins function, one of the ingredients that have
attracted recently much attention within the phenomenological studies of spin
asymmetries. Our analysis can be readily implemented to revisit previously
obtained fits that involve data at different scales for other spin-dependent
functions. Such improved fits are important to get better predictions --with
the correct evolution kernel-- for certain upcoming experiments aiming to
measure Sivers function, Collins function, transversity and other
spin-dependent functions as well.
