  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 T_c is limited by the spin-fluctuation scale $\Gamma$ and shows a
pronounced dependence on the next-nearest-neighbor hopping t'. The latter can
offer an explanation for the variation of T_c among different families of
cuprates.
