We present a comprehensive review of the discrete Boussinesq equations based
on their three-component forms on an elementary quadrilateral. These equa-
tions were originally found by Nijhoff et al using the direct linearization method
and later generalized by Hietarinta using a search method based on multidi-
mensional consistency. We derive from these three-component equations their
two- and one-component variants. From the one-component form we derive two
different semi-continuous limits as well as their fully continuous limits, which
turn out to be PDE’s for the regular, modified and Schwarzian Boussinesq
equations. Several kinds of Lax pairs are also provided. Finally we give their
Hirota bilinear forms and multi-soliton solutions in terms of Casoratians.