The basic theory of internal solitary waves is reviewed, with the main emphasis on applications
to the many observations of such waves in shallow coastal seas, fjords, lakes and in
the atmospheric boundary layer. Commencing with the equations of motion for an inviscid,
incompressible density-stratified fluid, we describe asymptotic reductions to model long-wave
equations, including the well-known Korteweg-de Vries equation, and several extensions. These
include a variable-coefficient extended Korteweg-de Vries equation which is proposed as an appropriate
evolution equation to describe internal solitary waves in environmental situations.
Various analytical and numerical solutions will be discussed.