The theory of superconductivity within the t-J model, as relevant for cuprates, is developed. It is based on the equations of motion for projected fermionic operators and the mode-coupling approximation for the self-energy matrix. The dynamical spin susceptibility at various doping is considered as an input, extracted from experiments. The analysis shows that the superconductivity onset is dominated by the spin-fluctuation contribution. We show that Tc is limited by the spin-fluctuation scale Γ and shows a pronounced dependence on the next-nearest-neighbor hopping t ′ . The latter can offer an explanation for the variation of Tc among different families of cuprates. PACS numbers: 71.27.+a, 74.20.Mn Since the discovery of high-temperature superconductivity (SC) in cuprates the mechanism of SC in these compounds represents one of the central open questions in the solid state theory. The role of strong correlations and the antiferromagnetic (AFM) state of the reference insulating undoped compound has been recognized very early [1]. Still, up to date there is no general consensus whether ingredients as embodied within the prototype single-band models of strongly correlated electrons are sufficient to explain the onset of high T c , or in addition other degrees of freedom, as e.g. phonons, should be invoked. As the basis of our study, we assume the simplest t-J model [2], allowing besides the nearest-neighbor (NN) hopping t also for the next-nearest-neighbor (NNN) hopping t ′ term. The latter model, as well as the Hubbard model [3], in the strong correlation limit U ≫ t closely related to the t-J model, have been considered by numerous authors to address the existence of SC due to strong correlations alone. Within the parent resonating-valence-bond (RVB) theory [1, 2] and slave-boson approaches to the t-J model [4] the SC emerges due to the condensation of singlet pairs, induced by the exchange interaction J. An alternative view on strong correlations has been that AFM spin fluctuations, becoming particularly longer-ranged and soft at low hole doping, represent the relevant low-energy bosonic excitations mediating the attractive interaction between quasiparticles (QP) and induce the d-wave SC pairing. The latter scenario has been mainly followed in the planar Hubbard model [3] and in the phenomenological spin-fermion model [5]. Recent numerical studies of the planar t-J model using the variational quantum Monte Carlo approach [6], as well as of the Hubbard model using cluster dynamical mean-field approximation [7], seem to confirm the stability of the d-wave SC as the ground state at intermediate hole doping.The t-J model is nonperturbative by construction, so it is hard to design for it a trustworthy analytical method. One approach is to use the method of equations of motion (EQM) to derive an effective coupling between fermionic QP and spin fluctuations [8]. The latter method has been employed to evaluate the self energy and anomalous properties of the spectral function [8,9,10], in particular the appearance of ...