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We present a review of theoretical investigations into the Kohn-Luttingernonphonon superconductivity mechanism in various 3D and 2D repulsive electronsystems described by the Fermi-gas, Hubbard, and Shubin-Vonsovsky models. Phasediagrams of the superconducting state are considered, including regions ofanomalous $s$-, $p$-, and $d$-wave pairing. The possibility of a strongincrease in the superconducting transition temperature $T_c$ even for a lowelectron density is demonstrated by analyzing the spin-polarized case or thetwo-band situation. The Kohn-Luttinger theory explains or predictssuperconductivity in various materials such as heterostructures and semimetals,superlattices and dichalcogenides, high-$T_c$ superconductors and heavy-fermionsystems, layered organic superconductors, and ultracold Fermi gases in magnetictraps. This theory also describes the anomalous electron transport and peculiarpolaron effects in the normal state of these systems. The theory can be usefulfor explaining the origin of superconductivity and orbital currents (chiralanomaly) in systems with the Dirac spectrum of electrons, including superfluid$^3$He-A, doped graphene, and topological superconductors.

The Kohn-Luttinger effect and anomalous pairing in new superconducting systems and graphene