In this paper, we propose a computational fluid-structure interaction (FSI) framework for the simulations of the interaction between free-surface flow and floating structures, such as offshore wind turbines. The framework is based on a suitable combination of the finite element method (FEM) and isogeometric analysis (IGA), and has good efficiency, accuracy and robustness characteristics. The free-surface phenomena are modeled using the Navier-Stokes equations of incompressible two-fluid flow in conjunction with the level set method. The FEM-based moving-domain ALE-VMS technique is employed to discretize the fluid mechanics equations, while the IGA-based rotation-free shell and beam/cable formulation is employed to model the mechanics of floating structures. The kinematic and traction compatibility at the fluid structure interface is handled by means of a recently-developed augmented Lagrangian FSI formulation with formal elimination of the Lagrange multiplier variable. A quasi-direct coupling strategy is adopted to handle the large added mass, and implemented by means of a matrix-free technique. The mathematical formulation of the FSI problem and its algorithmic implementation are described in detail, and two numerical test cases are presented. The first case is a free-surface-flow benchmark example of a solitary wave impacting a fixed, rigid platform. The second case is a set of full-scale free-surface-FSI simulations of the OC3-Hywind floating wind turbine design subjected to wave action. The computational results are compared with experimental and simulation data, with good agreement achieved in all cases where such data was available. Wind-turbine computations in the regime of high-amplitude waves are also presented. (C) 2016 Elsevier Ltd. All rights reserved.