The contractile ring is a remarkable tension-generating cellular machine that constricts and divides cells into two during cytokinesis, the final stage of the cell cycle. Since the ring's discovery, the parallels with muscle have been emphasized. Both are contractile actomyosin machineries, and long ago, a muscle-like sliding filament mechanism was proposed for the ring. This review focuses on the mechanisms that generate ring tension and constrict contractile rings. The emphasis is on fission yeast, whose contractile ring is sufficiently well characterized that realistic mathematical models are feasible, and possible lessons from fission yeast that may apply to animal cells are discussed. Recent discoveries relevant to the organization in fission yeast rings suggest a stochastic steady-state version of the classic sliding filament mechanism for tension. The importance of different modes of anchoring for tension production and for organizational stability of constricting rings is discussed. Possible mechanisms are discussed that set the constriction rate and enable the contractile ring to meet the technical challenge of maintaining structural integrity and tension-generating capacity while continuously disassembling throughout constriction.