The capability to rapidly and successfully move into new business models is an important source of sustainable competitive advantage and a key leverage to improve the sustainability performance of organisations. However, research suggests that many business model innovations fail. Despite the importance of the topic, the reasons for failure are relatively unexplored, and there is no comprehensive review of the sustainable business model innovation literature. This research provides a review of the literature, using a systematic database search and cross-reference snowballing. Its key contributions are: (1) a review of the key underlying concepts, discussing their similarities and differences and offer new definitions where there is an identified need; (2) we identify a research gap; and (3) we deduct research questions to address the gap. •Business model innovation is key leverage for financial and sustainability performance.•Many business model innovation attempts fail, this remains relatively unexplored.•We provide a comprehensive literature review of sustainable business model innovation.•We compared and defined key underlying concepts.•We identified a research gap and proposed research questions.