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We develop a theory based on the formalism of quasiclassical Green'sfunctions to study the spin dynamics in superfluid $^3$He. First, we derivekinetic equations for the spin-dependent distribution function in the bulksuperfluid reproducing the results obtained earlier without quasiclassicalapproximation. Then we consider a spin dynamics near the surface of fullygapped $^3$He-B phase taking into account spin relaxation due to thetransitions in the spectrum of localized fermionic states. The lifetime oflongitudinal and transverse spin waves is calculate taking into account theFermi-liquid corrections which lead to the crucial modification of fermionicspectrum and spin responses.

Quasiclassical theory of spin dynamics in superfluid $^3$He: kinetic equations in the bulk and spin response of surface Majorana states