Floating wind turbines are complex dynamic systems. The dynamics and design of such systems poses interesting engineering and scientific challenges. Although there already exist some informative and valuable reviews of this topic, there has been a lot of additional and important studies in more recent years. This topical paper reviews the main developments and research avenues currently under way, focusing on recent research literature. It adresses specific challenges of floating wind turbines and introduces the state-of-the-art and key issues focussing on the floating platform. In addition, recommendations for further research are introduced.