Internal waves are one of the most important phenomena
in geophysical fluid dynamics. Surprisingly, this type of
wave motion seems to be only partially understood even
within the relatively well controlled set-up of wave tank
experiments. This talk will discuss the fundamental dynamical
assumptions which lead to known models of internal wave
propagation in two-fluid systems, and will review the comparisons
of these models with available experimental data. It will be shown that
the
asymptotic expansions based on the weak nonlinearity assumption
are inadequate for the majority of dynamical regimes.
A derivation of a class of new models to remedy this
situation will be sketched.  The resulting evolution equations retain
the simplicity of known models while striving to
maintain the full nonlinearity of the original Euler equations. The
general structure of the new models encompasses known weakly
nonlinear theories,  such as those of Korteweg-de
Vries (KdV) and Benjamin-Ono (BO), while presenting new
mathematical challenges in the theory of  evolution
equations. A surprising spin-off of this modelling effort,
under appropriate restrictions, is a class
of completely integrable equations that support a rich
variety of dynamical regimes, from the classical KdV solitons
to particles on a lattice interacting through long range
forces.