There are only 3 methods for the production of heavy and superheavy (SH) nuclei, namely, a sequence of neutron capture and beta(−) decay, fusion reactions, and multinucleon transfer reactions. A macroscopic amount of the long-living SH nuclei located at the island of stability may be produced by using the pulsed nuclear reactors of the next generation only if the neutron fluence per pulse will be increased by about three orders of magnitude. Low values of the fusion cross sections and very short half-lives of nuclei with Z > 120 also put obstacles in synthesis of new elements. At the same time, an important area of SH isotopes located between those produced in the cold and hot fusion reactions remains unstudied yet. This gap could be filled in fusion reactions of 48 Ca with available lighter isotopes of Pu, Am, and Cm. New neutron-enriched isotopes of SH elements may be produced with the use of a 48 Ca beam if a 250 Cm target would be prepared. In this case, we get a real chance to reach the island of stability owing to a possible beta(+) decay of 291 114 and 287 112 nuclei formed in this reaction with a cross section of about 0.8 pb. Multinucleon transfer processes look quite promising for the production and study of neutron-rich heavy nuclei located in the upper part of the nuclear map not reachable by other reaction mechanisms. 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 allows one to plan such experiments at currently available accelerators.