The tension mooring of the floating ocean structures seems promising to expand the ocean space exploitation. This kind of mooring method becomes relatively inexpensive in deep ocean, and has many advantages such that it can suppress the motion of the floating structures and needs only minimum mooring space. This paper presents the results of the fundamental study on the tension mooring. A semisubmersible model with 4 footing columns is tautly moored by the rubber-chain cable in the regular waves. The results of the study are following. (1) The rubber-chain cable has a nonlinear load-elongation characteristics. The static characteristics before stress relaxation is applicable to the dynamic analysis. (2) The heaving and pitching motions of the floating structures are negligible, but the surging motion increases as the wave length does. (3) The frequency responses of the tension fluctuations of the cable do not depend on the material in case of the large initial tension. The axial rigidity of the rubber-chain cable is, however, much less than that of the wire cable so that in designing the rubber-chain tension mooring system, we must carefully set the natural frequency of the system sufficiently out of the wave spectrum. (4) The snap load easily occurs when the initial tension is small. (5) The analysis model is applicable to the design of the tension mooring system. It predicts, however, much larger snap load. The authors expect that the analysis model will be improved by including the nonlinear hydrodynamic forces.