Transitioning to a low‐carbon economy will entail sweeping transformations of energy and economic systems. A growing research body has raised concerns about the effect of such strain on financial stability. This literature on “financial transition risk” has highlighted that the conjunction of climate policy, technological change and shifts in consumption patterns may propagate to financial markets. In extreme cases, these dynamics may result in a “Climate‐Minsky” moment with systemic implications. The field has developed quickly, covering many methods and research questions. While this expansion in literature is advantageous when studying a complex issue like the low‐carbon transition, it also comes with downsides. The large number of methods hampers result comparison, and the integration of research designs. It also makes it difficult to provide a synthetic view of results in the literature as well as identify remaining uncertainties. To bridge these gaps, I propose a critical review of the literature. I examine three sub‐fields: the asset stranding literature, the direct assessment of transition risks through prospective models and the financial empirics of the low‐carbon transition. I expound their main results, critically assess underlying methodologies and propose a framework to compare results. The review ends by suggesting some avenues for future research.