Within the framework of the t-J model, we have calculated the temperature d
ependence of the nuclear spin-lattice relaxation rate, 1/T-1, of Cu-63, O-1
7, and Y-89 in the normal state of the superconductor YBa2Cu3O7. The main c
ontribution to the Cu rate arises from strong short-range antiferromagnetic
correlations between copper spins; the O-17 and Y-89 rates are dominated b
y spin fluctuations of longer wavelength. The theory is able to reproduce t
he main features of the temperature dependence of 1/T-1 as observed experim
entally.