  We present a systematic phenomenological description of Gilbert damping in
two-sublattice magnets. Our theory covers the full range of materials from
ferro- via ferri- to antiferromagnets. Following a Rayleigh dissipation
functional approach within a Lagrangian classical field formulation, the theory
captures intra- as well as cross-sublattice terms in the Gilbert damping,
parameterized by a 2$\times$2 matrix. When spin-pumping into an adjacent
conductor causes dissipation, we obtain the corresponding Gilbert damping
matrix in terms of the interfacial spin-mixing conductances. Our model
reproduces the experimentally observed enhancement of the ferromagnetic
resonance linewidth in a ferrimagnet close to its compensation temperature
without requiring an increased Gilbert parameter. It also predicts new
contributions to damping in an antiferromagnet and suggests the resonance
linewidths as a direct probe of the sublattice asymmetry, which may stem from
boundary or bulk.
