Problems of production and study of new neutron-enriched heavy nuclei are discussed. Low-energy multinucleon transfer reactions are shown to be quite appropriate for this purpose. Reactions with actinide beams and targets are of special interest for synthesis of new neutron-enriched transfermium nuclei and not-yet-known nuclei with closed neutron shell $N=126$ having the largest impact on the astrophysical r-process. The estimated cross sections for the production of these nuclei look very promising for planning such experiments at currently available accelerators. These experiments, however, are rather expensive and difficult to perform because of low intensities of the massive projectile beams and problems of separating and detecting the heavy reaction products. Thus, realistic predictions of the corresponding cross sections for different projectile-target combinations are definitely required. Some uncertainty still remains in the values of several parameters used for describing the low-energy nuclear dynamics. This uncertainty does not allow one to perform very accurate predictions for the productions of new heavier-than-target (trans-target) nuclei in multinucleon transfer reactions. Most of these parameters (nucleon transfer rate, nuclear viscosity, and fission barriers) are fundamental characteristics of low-energy nuclear dynamics. Determination of the values of these parameters (as well as their temperature dependence) is of significance in its own right. The available experimental data on the production of heavy nuclei in low-energy multinucleon transfer reactions are still insufficient and fragmentary. Several new experiments are proposed, these include those in which the role of shell effects in reaction dynamics can be better clarified.

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