  In this paper, we review the history and current state-of-the-art in the
modelling of long nonlinear dispersive waves. For the sake of conciseness of
this review, we omit the unidirectional models and focus especially on some
classical and improved Boussinesq-type and Serre-Green-Naghdi equations.
Finally, we propose also a unified modelling framework which incorporates
several well-known and some less known dispersive wave models. The present
manuscript is the first part of a series of two papers. The second part will be
devoted to the numerical discretization of a practically important model on
moving adaptive grids.
