Forty-fifth anniversary of the Nambu-Jona-Lasinio model

A short review of the development of the Nambu-Jona-Lasinio (NJL) model is given. The SU(2)x SU(2) and U(3)x U(3) local quark NJL models are considered. The mechanisms of spontaneous breaking of chiral symmetry and vector dominance are shown. The local NJL model allows us to describe the mass spectrum and main strong and electroweak decays of the four ground-state nonets of pseudoscalar, scalar, vector and axial-vector mesons. Applications of this model to the description of mesons in hot and dense medium are discussed. It is shown that for solving problems connected with the description of the meson radial excitations and quark confinement it is necessary to consider a nonlocal extension of the NJL model. The main attention is attracted to the description of the methods used in different versions of the NJL model. Physical results for low-energy hadron physics obtained in these models can be found in the cited works.