The multimodal method reduces the sloshing problem with free surface to a (modal) system of nonlinear ordinary differential equations. The method was originally proposed for nonimpulsive hydrodynamic loads. However, recently it has been successfully extended to the case of sloshing-induced slamming. In the 1950–1960’s, this method was used in the computational fluid dynamics (CFD) but later was replaced by the algorithms developed in the 1990-2000’s. At present, the method plays a dual role: first , as a unique analytic tool for the investigation of nonlinear sloshing regimes, their stability, and chaos as well as for simulations when traditional CFD fails (e.g., in the case of containers with perforated screen) and, second , as a source of construction of the modal systems, which are analogs of the Korteweg–de-Vries, Boussinesq, and other equations but for bounded volumes of liquid. We present a survey of the state-ofthe-art of the problem, describe the existing modal systems, and formulate open problems.
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