In magnetization dynamics, the Gilbert damping α is often taken as a parameter. We report on a theoretical investigation of α, taking into account crystal symmetries, spin-orbit coupling and thermal reservoirs. The tensor α is calculated within the Kambersky breathing Fermi-surface model. The computations are performed within a tight-binding electronic structure approach for the bulk and semi-infinite systems. Slater-Koster parameters are obtained by fitting the electronic structure to first-principles results obtained within the multiple-scattering theory. We address the damping tensor for the bulk and surfaces of the transition metals Fe and Co. The role of various contributions are investigated: intra- and interband transitions, electron and magnetic temperature as well as surface orientation. Our results reveal a complicated non-local, anisotropic damping that depends on all three thermal reservoirs.