In this paper, NMPC schemes based on fast update methods (fast NMPC schemes) are reviewed that strive to provide a fast but typically suboptimal update of the control variables at each sampling instant with negligible computational delay. The review focuses on schemes that employ one of two subclasses of fast update methods developed for direct solution approaches, the suboptimal update methods and the sensitivity-based update methods. The connections and similarities of the fast update methods, the elements of the fast NMPC, the control architecture as well as the fast NMPC schemes as a whole are highlighted to support the assessment of the benefits and limitations of each individual scheme. In this way, this review facilitates the choice of a suitable fast NMPC scheme within the vast amount of fast NMPC schemes available in literature. (C) 2016 Elsevier Ltd. All rights reserved.