  In the present article we consider the problem of wave interaction with a
partially immersed, but floating body. We assume that the motion of the body is
prescribed. The general mathematical formulation for this problem is presented
in the framework of a hierarchy of mathematical models. Namely, in this first
part we formulate the problem at every hierarchical level. The special
attention is payed to fully nonlinear and weakly dispersive models since they
are most likely to be used in practice. For this model we have to consider
separately the inner (under the body) and outer domains. Various approached to
the gluing of solutions at the boundary is discussed as well. We propose
several strategies which ensure the global conservation or continuity of some
important physical quantities.
