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Predictions of the wave-induced response of floating structures that are moored in a harbour or coastal waters require an accurate description of the (nonlinear) evolution of waves over variable bottom topography, the interactions of the waves with the structure, and the dynamics of the mooring system. In this paper, we present a new advanced numerical model to simulate the wave-induced response of a floating structure that is moored in an arbitrary nearshore region. The model is based on the non-hydrostatic approach, and implemented in the open-source model SWASH, which provides an efficient numerical framework to simulate the nonlinear wave evolution over variable bottom topographies. The model is extended with a solution to the rigid body that is tightly coupled to the hydrodynamic equations. The model was validated for two test cases that consider different floating structures of increasing geometrical complexity: a cylindrical geometry that is representative of a wave-energy-converter, and a vessel with a more complex shaped hull. A range of wave conditions were considered, varying from monochromatic to short-crested sea states. Model predictions of the excitation forces, added mass, radiation damping, and the wave-induced response agreed well with benchmark solutions to the potential flow equations. Besides the response to the primary wave (sea-swell) components, the model was also able to capture the second-order difference-frequency forcing and response of the moored vessel. Importantly, the model captured the wave-induced response with a relatively coarse vertical resolution, allowing for applications at the scale of a realistic harbour or coastal region. The proposed model thereby provides a new tool to seamlessly simulate the nonlinear evolution of waves over complex bottom topography and the wave-induced response of a floating structure that is moored in coastal waters.