The present paper is dedicated to the development of a numerical model for the water impact of two-dimensional (2D) and axisymmetric bodies with imposed motion. The work is a first step towards the implementation of a 2D+t procedure to be used for the analysis of aircraft ditching. The problem is investigated under the assumptions of an inviscid and incompressible fluid, which is modeled by a potential flow model with fully non-linear boundary conditions at the free-surface. The unsteady boundary value problem with a free-surface is numerically solved through a boundary element method, coupled to a simplified finite element method to describe the thinnest part of the jet. The study is aimed at describing the entry and exit phases. Specific numerical solutions are developed to tackle the exit phase and to improve the stability of the model. Results are presented in terms of free-surface shape, pressure distribution and hydrodynamic load acting on the impacting body. The model is used to study the water entry and exit of a 2D wedge and an axisymmetric cone, for which numerical or experimental results are available in the literature. The numerical investigation shows that the proposed model accurately simulate both the entry and exit phases. For the exit phase, it is shown that the proposed model, being fully non-linear, provides a much better prediction of the loads and the wetted area compared to simplified (analytical) approaches. The effects of the gravity, usually missing in the approaches available in the literature, are also investigated, showing they are rather important, especially, in the exit phase.