Structural symmetry breaking in magnetic materials is responsible for the existence of multiferroics, current-induced spin-orbit torques and some topological magnetic structures. In this Letter we report that the structural inversion asymmetry (SIA) gives rise to a chiral damping mechanism, which is evidenced by measuring the field-driven domain-wall (DW) motion in perpendicularly magnetized asymmetric Pt/Co/Pt trilayers. The DW dynamics associated with the chiral damping and those with Dzyaloshinskii-Moriya interaction (DMI) exhibit identical spatial symmetry. However, both scenarios are differentiated by their time reversal properties: whereas DMI is a conservative effect that can be modelled by an effective field, the chiral damping is purely dissipative and has no influence on the equilibrium magnetic texture. When the DW motion is modulated by an in-plane magnetic field, it reveals the structure of the internal fields experienced by the DWs, allowing one to distinguish the physical mechanism. The chiral damping enriches the spectrum of physical phenomena engendered by the SIA, and is essential for conceiving DW and skyrmion devices owing to its coexistence with DMI (ref. ).