The review discusses the emergence of the spin-fermion model of cuprates and formation of the spin-polaron concept of the electronic structure of hole-doped cuprate superconductors. This concept allowed to describe the properties of cuprates in the normal phase as well as the features of the superconducting pairing in the unified approach. The obtaining of the spin-fermion model from the Emery model in the regime of strong electronic correlations is described in detail and it is shown how the effects of the covalent mixing of atomic states induce the strong interplay between the spins of copper ions and holes on oxygen ions. Such a strong interaction on the singlet state of the spin subsystem of copper ions (quantum spin liquid) background leads to the formation of special Fermi quasiparticles — nonlocal spin polarons. When doping, the spin-polaron ensemble exhibits instability toward the superconducting d-wave pairing, whereas the superconducting s-wave pairing is not implemented. At the optimal doping, the transition to the superconducting phase occurs at temperatures corresponding to experimental data. It is shown that the superconducting d-wave pairing of the spin-polaron quasiparticles is not suppressed by the Coulomb repulsion of the holes located on neighboring oxygen ions. It is emphasized that at the account for the spectral characteristics of spin-polaron quasiparticles, the calculated temperature and doping dependences of the London penetration depth are in a good agreement with experimental data.