European Journal of Neuroscience published by Federation of European Neuroscience Societies and John Wiley & Sons Ltd.Within the reductionist framework, researchers in the special sciences formulate key terms and concepts and try to explain them with lower-level science terms and concepts. For example, behavioural vision scientists describe contrast perception with a psychometric function, in which the perceived brightness increases logarithmically with the physical contrast of a light patch (the Weber-Fechner law). Visual neuroscientists describe the output of neural circuits with neurometric functions. Intuitively, the key terms from two adjacent scientific domains should map onto each other; for instance, psychometric and neurometric functions may map onto each other. Identifying such mappings has been the very goal of neuroscience for nearly two centuries. Yet mapping behaviour to brain measures has turned out to be difficult. Here, we provide various arguments as to why the conspicuous lack of robust brain–behaviour mappings is rather a rule than an exception. First, we provide an overview of methodological and conceptual issues that may stand in the way of successful brain–behaviour mapping. Second, extending previous theoretical work (Herzog, Doerig and Sachse, 2023), we show that brain–behaviour mapping may be limited by complexity barriers. In this case, reduction may be impossible.