  A relaxation-function theory for the dynamic spin susceptibility in the
$t$--$J$ model is presented. By a sum-rule-conserving generalized mean-field
approximation (GMFA), the two-spin correlation functions of arbitrary range,
the staggered magnetization, the uniform static susceptibility, and the
antiferromagnetic correlation length are calculated in a wide region of hole
doping and temperaturs. A good agreement with available exact diagonalization
(ED) data is found. The correlation length is in reasonable agreement with
neutron-scattering experiments on La_{2-\delta}Sr_\delta)CuO_4. Going beyond
the GMFA, the self-energy is calculated in the mode-coupling approximation. The
spin dynamics at arbitrary frequencies and wave vectors is studied for various
temperatures and hole doping. At low doping a spin-wave-type behavior is found
as in the Heisenberg model, while at higher doping a strong damping caused by
hole hopping occurs, and a relaxation-type spin dynamics is observed in
agreement with the ED results. The local spin susceptibility and its (\omega/T)
scaling behavior are calculated in a reasonable agreement with experimental and
ED data.
