The motivation for this study is the fast development of floating offshore wind energy and the immature methodology and engineering practice related to predictions of vortex-induced motions (VIM). Benefiting from the oil and gas industry, in the past several decades, extensive knowledge and experience on vortex-induced vibrations (VIV) on slender marine structures has been gained. As the learnings from these efforts should be transferred and adapted to the renewable energy industry, a state-of-the-art review on influential VIM research has been carried out in this paper, focusing on: (1) engineering practice, (2) model tests, (3) numerical calculation, and (4) field measurement. Engineering gaps and potential research topics are identified as future work.