  The motion of both internal and surface waves in incompressible fluids under
capillary and gravity forces is a major research topic. In particular, we
review the derivation of some new models describing the dynamics of
gravity-capillary nonlinear waves in incompressible flows. These models take
the form of both bidirectional and unidirectional nonlinear and nonlocal wave
equations. More precisely, with the goal of telling a more complete story, in
this paper we present the results in the works
\cite{GS,aurther2019rigorous,granero2019models,granero2019well,granero2020well,granero2021motion,
granero2021global} together with some new results regarding the well-posedness
of the resulting PDEs.
