The problem of describing the emergence of sea waves on the shore, obviously, is of great practical importance. At the same time, this is a rather difficult task due to the effects of strong nonlinearity and wave breaking. The greatest progress has been made in the framework of the nonlinear theory of shallow water for specific geometries of the coastal zone. There is a set of exact solutions used as test solutions to control numerical calculations. The report will give an overview of the problem of wave run-up on the shore from a mathematical point of view and those restrictions imposed from a geophysical point of view. The following questions will be considered:
- long wave run-up on a flat slope,
- wave run-up in narrow inclined channels (bays),
- collapse condition,
- run-up of irregular (random waves) on the shore,
- effects of dispersion and dissipation.
