  Employing the stochastic mean-field (SMF) approach, we develop a quantal
diffusion description of the multi-nucleon transfer in heavy-ion collisions at
finite impact parameters. The quantal transport coefficients are determined by
the occupied single-particle wave functions of the time-dependent Hartree-Fock
equations. As a result, the primary fragment mass and charge distribution
functions are determined entirely in terms of the mean-field properties. This
powerful description does not involve any adjustable parameter, includes the
effects of shell structure and is consistent with the fluctuation-dissipation
theorem of the non-equilibrium statistical mechanics. As a first application of
the approach, we analyze the fragment mass distribution in
$^{48}\mathrm{Ca}+{}^{238}\mathrm{U}$ collisions at the bombarding energy
$E_{\text{c.m.}}=193$ MeV and compare the calculations with the experimental
data.
