A two-degree-of-freedom model of offshore structures is studied for its dynamic response in a steady-current and regular-wave environment. The two degrees of freedom represent motions in-line with, and orthogonal to, the current direction. The hydrodynamic forces on the structure are modeled by the modified Morison equation and, additionally, a non-linear symmetric hardening spring is included in the model. For "large inertia" structures, both the cases of coincident and non-coincident current and wave directions are investigated. By using the shooting technique, the steady state periodic response is numerically obtained over a wide range of wave frequencies. Floquet theory is utilized to evaluate the stability of these solutions. Non-periodic steady state solutions, arising due to instabilities of the periodic responses, are obtained by direct time integration. For large damping and drag coefficients and for coincident current and wave directions, the only response possible is in the direction of the current. As the wave direction deviates from the current direction, the response becomes bidirectional, and both the in-line and the orthogonal components exhibit primary and superharmonic resonances. When the wave is orthogonal to the current, the primary resonance in the current direction is completely eliminated. For lower damping and drag coefficients, a bidirectional primary response is excited even for the case of coincident wave and current directions. As the damping is decreased the periodic bidirectional motion is shown to further bifurcate to periodic, and then chaotic, amplitude-modulated motions. The chaotic motion is, over an interval in wave frequencies, found to be destroyed by heteroclinic bifurcations.