The main subject of this thesis is to show that transfer reactions with the neutron-rich unstable beams and the heavy targets could be very adequate for the production of the neutron-rich heavy nuclei. To that purpose, the measurement of 94^Rb+208^Pb transfer reaction has been performed at the energy of about 30% above the Coulomb barrier, provided by the radioactive ion beam facility of CERN (HIE-ISOLDE). The high-resolution and high-efficiency experimental set-up consisted of a large γ-array MINIBALL, coupled to the CD fragment detector of wide angular range. From the measured particle-γ and particle-γ-γ coincidences, the characteristic γ transitions were attributed to 207^Pb, 208^Pb, 209^Pb and 210^Pb, and their binary partners, the isotopes of Rb. Based on the attributed γ transitions, the total cross sections in Pb isotopes were successfully constructed. In particular, in channels populated by transfer of neutrons from 94^Rb beam to 208^Pb target, the cross section of 138(8) mb was evaluated in 209^Pb, and 41(7) mb in 210^Pb. These significant values quantify the production of the moderately neutron-rich heavy nuclei for the first time in the transfer reaction induced by an unstable beam. The obtained results were fairly reproduced with the semi-classical model GRAZING, thus encouraging the use of the presented approach for the population of the neutron-rich heavy nuclei. This thesis strongly suggests that the presently available experimental techniques and radioactive ion beam intensities are very suitable to extend the reaction mechanism and the nuclear structure studies farther from the valley of stability.