A two-dimensional Boundary Element Method (BEM) is developed to study the strongly nonlinear interaction between a surface-piercing body and the free-surface. The scheme is applied to problems with and without the possibility of ventilation resulting from the motion and geometric configuration of the surface-piercing body. The main emphasis of this research work is on the development of numerical methods to improve the performance prediction of surface-piercing propellers by including the whole range of free-surface nonlinearities. The scheme is applied to predict the ventilated cavity shapes resulting from the vertical and rotational motion of a blade-section with fully nonlinear free-surface boundary conditions. The current method is able to predict the ventilated cavity shapes for a wide range of angles of attack and Froude numbers, and is in good agreement with existing experimental results. Through a comparison with a linearized free-surface method, the current method highlights the shortcomings of the negative image approach used commonly in two-dimensional and three-dimensional numerical methods for surface-piercing hydrofoils or propellers. The current method with all its capabilities makes it a unique contribution to improving numerical tools for the performance prediction of surface-piercing propellers. The scheme is also applied to predict the roll and heave dynamics of two-dimensional Floating Production Storage and Offloading (FPSO) vessel hull sections within a potential flow framework. The development of the potential flow model is aimed at validating the free-surface dynamics of an independently developed Navier Stokes Solver for predicting the roll characteristics of two-dimensional hull sections with bilge keels. / text