  We develop a theory based on the formalism of quasiclassical Green's
functions to study the spin dynamics in superfluid $^3$He. First, we derive
kinetic equations for the spin-dependent distribution function in the bulk
superfluid reproducing the results obtained earlier without quasiclassical
approximation. Then we consider a spin dynamics near the surface of fully
gapped $^3$He-B phase taking into account spin relaxation due to the
transitions in the spectrum of localized fermionic states. The lifetime of
longitudinal and transverse spin waves is calculate taking into account the
Fermi-liquid corrections which lead to the crucial modification of fermionic
spectrum and spin responses.
