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Wave, by definition, is an observable disturbance that travels through a medium from one location to another transferring energy. According to the properties of the medium, or the geometry, or the external forces, there might occur different circumstances and waves may behave in different manners. In this talk, we will briefly explain how the literature developed for water wave theory in that sense. Moreover, we will discuss nonlinear behaviors such as wave breaking followed by some examples. Some recent wave models will also be introduced and the questions that need to be answered will be addressed.