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Low values of the fusion cross sections and very short half-lives of nucleiwith Z$>$120 put obstacles in synthesis of new elements. Different nuclearreactions (fusion of stable and radioactive nuclei, multi-nucleon transfers andneutron capture), which could be used for the production of new isotopes ofsuperheavy (SH) elements, are discussed in the paper. The gap of unknown SHnuclei, located between the isotopes which were produced earlier in the coldand hot fusion reactions, can be filled in fusion reactions of $^{48}$Ca withavailable lighter isotopes of Pu, Am, and Cm. Cross sections for the productionof these nuclei are predicted to be rather large, and the correspondingexperiments can be easily performed at existing facilities. For the first time,a narrow pathway is found to the middle of the island of stability owing topossible $\beta^+$-decay of SH isotopes which can be formed in ordinary fusionreactions of stable nuclei. Multi-nucleon transfer processes at near barriercollisions of heavy (and very heavy, U-like) ions are shown to be quiterealistic reaction mechanism allowing us to produce new neutron enriched heavynuclei located in the unexplored upper part of the nuclear map. Neutron capturereactions can be also used for the production of the long-living neutron richSH nuclei. Strong neutron fluxes might be provided by pulsed nuclear reactorsand by nuclear explosions in laboratory conditions and by supernova explosionsin nature. All these possibilities are discussed in the paper.

Future of superheavy element research: Which nuclei could be synthesized within the next few years?