The nonlinear vibrations and random vibrations of the offshore floating structures moored by cables are studied in both two dimensions (2D) and three dimensions (3D). The nonlinear equations of motions of the mooring cables are derived using the 2D or 3D cable elements which are formulated based on the extended Hamilton principle. The floating structure is simplified as a rigid body with three degrees of freedom in two-dimensions and six degrees of freedom in three-dimensions. The added mass effect and hydrodynamic forces such as viscous drag force and Froude-Krylov force that act on both the mooring cables and floating platform are taken into consideration. The connection conditions between the mooring cables and floating platform are given to formulate the equations of motion of the whole structure. Finally, the equations of motion of the whole system are analyzed numerically under various kinds of loadings, such as deterministic sinusoidal excitation, seismic excitation and sea wave excitation. For the deterministic analysis, the response amplitudes of the cable-moored floating structure are figured out for different excitation frequencies of sinusoidal excitation. For the random vibration analysis, the probability density functions of the structural responses are studied. The influences of different sag-to-span ratios and inclination angles of the mooring cables are investigated on the responses or the statistical responses of the floating platform and those of the maximum tensile force in the cables.