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This paper reviews the many criticisms that Integrated Assessment Models (IAMs)the bedrock of mitigation analysishave received in recent years. Critics have asserted that there is a lack of transparency around model structures and input assumptions, a lack of credibility in those input assumptions that are made visible, an over-reliance on particular technologies and an inadequate representation of real-world policies and processes such as innovation and behaviour change. The paper then reviews the proposals and actions that follow from these criticisms, which fall into three broad categories: scrap the models and use other techniques to set out low-carbon futures; transform them by improving their representation of real-world processes and their transparency; and supplement them with other models and approaches. The article considers the implications of each proposal, through the particular lens of how it would explore the role of a key low-carbon technologybioenergy with carbon capture and storage (BECCS), to produce net negative emissions. The paper concludes that IAMs remain critically important in mitigation pathways analysis, because they can encompass a large number of technologies and policies in a consistent framework, but that they should increasingly be supplemented with other models and analytical approaches.