The review concerns the theoretical aspects of studying the role of Jahn-Teller interactions in quasi-two-dimensional cuprate antiferromagnets and HTSs. The analysis of the results which allow the establishment of relation between the quasi-two-dimensional character of the properties of underdoped cuprate HTSs in the normal and superconducting states and the Jahn-Teller (JT) nature of bivalent copper ions is made. It is shown that the combination of these two peculiarities brings into existence a pseudogap state. In underdoped cuprate HTSs with JT lattice distortions the quasi-two-dimensionality gives rise to two-dimensional local and quasilocal states of charge carriers. It manifests itself in a substantial temperature dependence of the number of components of localized and delocalized states of charge carriers and in a repeated dynamical reduction of the dimension of underdoped cuprate HTSs with decreasing temperature. Such HTSs with a doping concentration less than the optimal one in the most part of the phase diagram in both the normal and superconducting states exist in a quasi-two-dimensional state. This implies that the superconducting state of underdoped cuprate HTSs differs from the BCS state and by its properties is closer to that of the quasi-two-dimensional Beresinskii - Kosterlitz - Thouless superconductor without off-diagonal long-range order (ODLRO). It is shown that the difference consists above all in the mechanism of superconductivity. Although the strong JT electron-phonon interaction plays the key role and results in the formation of two-site JT polarons, the attraction between holes and such polarons and

the formation of superfluid two-site JT polarons with antiferromagnetic cores are caused by the compensation of Coulomb repulsion by the polaron energy shift. A supposition is discussed that in overdoped cuprate HTSs the superconducting state results from the establishment of ODLRO in the three-dimensional BCS model with a nonpersistent total number of charge carriers and nonzero quantum fluctuations of the number of charge carriers.

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