  A Kubo-Greenwood-like equation for the Gilbert damping parameter $\alpha$ is
presented that is based on the linear response formalism. Its implementation
using the fully relativistic Korringa-Kohn-Rostoker (KKR) band structure method
in combination with Coherent Potential Approximation (CPA) alloy theory allows
it to be applied to a wide range of situations. This is demonstrated with
results obtained for the bcc alloy system Fe$_x$Co$_{1-x}$ as well as for a
series of alloys of permalloy with 5d transition metals.
  To account for the thermal displacements of atoms as a scattering mechanism,
an alloy-analogy model is introduced. The corresponding calculations for Ni
correctly describe the rapid change of $\alpha$ when small amounts of
substitutional Cu are introduced.
