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Recent floods in the United States and Asia again highlighted their devastating effects, and without investments in adaptation, the future impact of floods will continue to increase. Key to making accurate flood-risk projections are assessments of how disaster-risk reduction (DRR) measures reduce risk and how much risk remains after adaptation. Current flood-risk-assessment models are ill-equipped to address this, as they assume a static adaptation path, implying that vulnerability will remain constant. We present a multi-disciplinary approach that integrates different types of adaptive behaviour of governments (proactive and reactive) and households (rational and boundedly rational) in a continental-scale risk-assessment framework for river flooding in the European Union. Our methodology demonstrates how flood risk and adaptation might develop, indicates how DRR policies can steer decisions towards optimal behaviour, and indicates how much residual risk remains that has to be covered by risk-transfer mechanisms. We find that the increase in flood risk due to climate change may be largely offset by adaptation decisions. Moreover, we illustrate that adaptation by households may be more influential for risk reduction than government protection in the short term. The results highlight the importance of integrating behavioural methods from social sciences with quantitative models from the natural sciences, as advocated by both fields.