A phenomenology of magnetic chiral damping is proposed in the context of magnetic materials lacking inversion symmetry breaking. We show that the magnetic damping tensor adopts a general form that accounts for a component linear in magnetization grad
ient in the form of Lifshitz invariants. We propose different microscopic mechanisms that can produce such a damping in ferromagnetic metals, among which spin pumping in the presence of anomalous Hall effect and an effective $s$-$d$ Dzyaloshinskii-Moriya antisymmetric exchange. The implication of this chiral damping in terms of domain wall motion is investigated in the flow and creep regimes. These predictions have major importance in the context of field- and current-driven texture motion in noncentrosymmetric (ferro-, ferri-, antiferro-)magnets, not limited to metals.
