The prediction of ship motions and quiescent periods, is of paramount importance for the maritime industry. The capability to predict these events sufficiently in advance has the potential to improve the safety and efficiency of several marine operations, such as landing and take-off on aircraft carriers, transfer of cargo, and mating operations between ships. Several models have been proposed in the literature for the prediction of ship motions and quiescent period. This work will review them by first grouping them into three main categories (i.e., physical, data-driven, and hybrid models) and then by detailing the most recent and relevant ones describing the advantages and disadvantages of each approach. Review concludes with the open problems and future perspectives of this important field of research.