  We study the enhancement of the ferromagnetic relaxation rate in thin films
due to the adjacent normal metal layers. Using linear response theory, we
derive the dissipative torque produced by the s-d exchange interaction at the
ferromagnet-normal metal interface. For a slow precession, the enhancement of
Gilbert damping constant is proportional to the square of the s-d exchange
constant times the zero-frequency limit of the frequency derivative of the
local dynamic spin susceptibility of the normal metal at the interface.
Electron-electron interactions increase the relaxation rate by the Stoner
factor squared. We attribute the large anisotropic enhancements of the
relaxation rate observed recently in multilayers containing palladium to this
mechanism. For free electrons, the present theory compares favorably with
recent spin-pumping result of Tserkovnyak et al. [Phys. Rev. Lett.
\textbf{88},117601 (2002)].
