In this paper two non-linear control laws for ships are derived by using a non-linear ship model which includes the hydrodynamic e ects due to time-varying speed and wave frequency. The non-linear ship model does not have the structural properties of symmetric inertia and positive damping at high speed, which are common assumptions, when designing non-linear ship tracking control systems. Backstepping control designs are used to circumvent the problem of a non-symmetrical inertia matrix. The ΓΏrst case is ship tracking with full actuation in surge, sway and yaw while the second case discusses an emergency situation where a ship is supposed to track a time-varying trajectory by using the bow thruster and one of the main propellers as actuation devices only. The application of speed-and wave-dependent models for ship control is a ΓΏrst step towards the design of a new generation of non-linear ship control systems that can operate under time-varying speed and wave frequency conditions. Previous work has often relied on the assumption of constant speed, and practical solutions have been based on switching between di erent control schemes at di erent speeds. Slowly varying environmental disturbances, such as ocean current, second-order wave-induced disturbances and wind forces, are compensated for by using adaptive backstepping. Global uniform asymptotic stability (GUAS) is proven for the fully actuated case by using a recent theorem for GUAS when backstepping with integral action, whereas asymptotic tracking of position and velocity are shown for the degraded control case. Computer simulations illustrate the performance and the robustness of the control laws for an o shore supply vessel.