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This survey presents new mathematical results in the theory of linear and nonlinear waves on the surface of a flotation liquid. A flotation liquid is a liquid on whose surface heavy particles are floating; the particles may consist of arbitrary materials or may be particles of frozen liquid.The first part of the article considers initial-and boundary-value problems in the theory, their solvability, and the behavior of the solutions over long periods. In the second part of the survey, theorems are proved on the existence of nonlinear standing waves within the framework of an exact physical model, and both internal and free waves are considered. Also, the fundamental equations for shallow flotation waves are derived and examined.

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