The climate crisis demands a strong response from policy-makers worldwide. The current global climate policy agenda requires technological change, innovation, labour markets and the financial system to be led towards an orderly and rapid low-carbon transition. Yet progress has been slow and incremental. Inadequacies of policy appraisal frameworks used worldwide may be significant contributors to the problem, as they frequently fail to adequately account for the dynamics of societal and technological change. Risks are underestimated, and the economic opportunities from innovation are generally not assessed in practice. Here, we identify root causes of those inadequacies and identify them to structural features of standard analysis frameworks. We use a review of theoretical principles of complexity science and the science of dynamical systems and formulate a generalisation of existing frameworks for policy analysis and the appraisal of outcomes of proposed policy strategies, to help better identify and frame situations of transformational change. We use the term “risk-opportunity analysis” to capture the generalised approach, in which conventional economic cost-benefit analysis is a special case. New guiding principles for policy-making during dynamic and transformational change are offered.