  We pursue the consequences of a theory in which the resonance peak observed
in inelastic neutron scattering (INS) experiments on underdoped and optimally
doped YBa$_2$Cu$_3$O$_{6+x}$ compounds arises from a spin-wave excitation. We
find that it is heavily damped, and thus almost not observable, in the normal
state, but becomes visible in the superconducting state due to the drastic
decrease in spin damping. We show that a spin-fermion model correctly describes
the temperature dependence of the peak position for YBa$_2$Cu$_3$O$_7$, as well
as the doping dependence of the peak position and of the integrated intensity.
We explain why no resonance peak has been observed in La$_{2-x}$Sr$_x$CuO$_4$,
and make several predictions concerning resonance peaks in other cuprate
superconductors.
