  We review here the derivation of many of the most important models that
appear in the literature (mainly in coastal oceanography) for the description
of waves in shallow water. We show that these models can be obtained using
various asymptotic expansions of the "turbulent" and non-hydrostatic terms that
appear in the equations that result from the vertical integration of the free
surface Euler equations. Among these models are the well-known nonlinear
shallow water (NSW), Boussinesq and Serre-Green-Naghdi (SGN) equations for
which we review several pending open problems. More recent models such as the
multi-layer NSW or SGN systems, as well as the Isobe-Kakinuma equations are
also reviewed under a unified formalism that should simplify comparisons. We
also comment on the scalar versions of the various shallow water systems which
can be used to describe unidirectional waves in horizontal dimension $d=1$;
among them are the KdV, BBM, Camassa-Holm and Whitham equations. Finally, we
show how to take vorticity effects into account in shallow water modeling, with
specific focus on the behavior of the turbulent terms. As examples of
challenges that go beyond the present scope of mathematical justification, we
review recent works using shallow water models with vorticity to describe wave
breaking, and also derive models for the propagation of shallow water waves
over strong currents.
