Broadband magnetic loss and damping behavior of Co-based amorphous ribbons and thin films have been investigated. The permeability and loss response of the transverse anisotropy ribbon samples in the frequency range DC to 1 GHz is interpreted in terms of combined and distinguishable contributions to the magnetization process by domain wall displacements and magnetization rotations. The latter alone are shown to survive at the highest frequencies, where the losses are calculated via coupled Maxwell and Landau-Lifshitz-Gilbert (LLG) equations. Remarkably high values of the LLG damping coefficient α = 0.1-0.2 are invoked in this theoretical prediction. Direct measurements of α by pulsed inductive microwave magnetometry are thus performed, both in these laminae and in amorphous films of identical composition, obtaining about one order of magnitude increase of the α value upon the 100 nm÷10 μm thickness range. This confirms that dissipation by eddy currents enters the LLG equation via large increase of the damping coefficient.