The luminosity performance of a future linear collider (LC) will depend critically on the performance of the damping rings. The design luminosities of the current LC proposals require rings with very short damping times, large acceptance, low equilibrium emittance and high beam intensity. We discuss the design strategies for lattices achieving the goals of dynamical stability, examine the challenges for alignment and coupling correction, and consider a variety of collective effects that threaten to limit beam quality. We put the design goals in context by referring to the experience of operating facilities, and outline the further research and development that is needed.