In the past decades, the damping constant α has been successfully described theoretically-in some cases even quantitatively-using various approaches such as the breathing Fermi-surface model 1,2 , the torque correlation model 3 , scattering theory 4,5 and the torquetorque correlation within a linear response model 6,7 . On the basis of these works, α is expected to scale as α ~ n(E F )ξ 2 τ −1 under certain circumstances, where n(E F ) is the density of states at the Fermi level E F , ξ is the strength of the spin-orbit interaction and τ is the electron momentum scattering time 8,9 . Indeed, the dependences on n(E F ) (refs 9-11 ), ξ (refs 12,13 ) and τ (ref.14 ) have been confirmed separately in a large variety of materials. In general, it is assumed that damping is isotropic. However, several theoretical works [15][16][17][18] have suggested that damping should be anisotropic in single-crystalline ferromagnetic metals, such as bulk Fe, Co and Ni. This prediction is based on the anisotropic electronic structure where the shape of the Fermi surface depends on the orientation of the magnetization direction due to the spin-orbit interaction. The anisotropic electronic structure and thus the anisotropic damping, however, can be dramatically reduced due to smearing of the energy bands in the presence of electron scattering, which makes the experimental observation of the anisotropic damping in bulk materials difficult. So far, only a few experiments [19][20][21][22] have tried to prove the existence of anisotropic damping in bulk magnets but convincing experimental evidence is still lacking.Here, we report the observation of anisotropic Gilbert damping in a quasi-two-dimensional Fe/GaAs(001) system. The idea behind this is to explore the interfacial spin-orbit interaction of a singlecrystalline ferromagnetic metal/semiconductor interface. Our findings differ distinctly from the theoretical predictions made for bulk magnets. The Fe/GaAs heterostructure was intensively studied in the past two decades for semiconductor spintronics, and has been utilized, for example, to realize spin injection at room temperature 23 . Recently, interest in this system has been revived in view of spin-orbit electronics, because of the existence of robust spin-orbit fields at the Fe/GaAs interface, which can cause a mutual conversion between spin and charge currents at room temperature 24 . The spinorbit fields, including both Bychkov-Rashba-and Dresselhaus-like terms, result from the C 2v symmetry of the interface 25 . Specifically, at the Fe/GaAs(001) interface, Fe Bloch states near E F penetrate into GaAs. Therefore, electrons of Fe 'feel' both Bychkov-Rashba and Dresselhaus spin-orbit interaction at the interface, causing a rich variety of interfacial spin-orbit-related phenomena. It has been found, for example, that the symmetry of anisotropic magnetoresistance 26 and the polar magneto-optic Kerr effect 27 of Fe is governed by the twofold interfacial C 2v symmetry rather than its bulk fourfold C 4v symmetry when the thickne...