concepts play a central role in human behaviour and constitute a critical component of the human conceptual system. Here, we investigate the neural basis of four types of abstract concepts, examining their similarities and differences through neuroimaging meta-analyses. We examine numerical and emotional concepts, and two higher-order abstract processes, morality judgements and theory of mind. Three main findings emerge. First, representation of abstract concepts is more widespread than is often assumed. Second, representations of different types of abstract concepts differ in important respects. Each of the domains examined here was associated with some unique areas. Third, some areas were commonly activated across domains and included inferior parietal, posterior cingulate and medial prefrontal cortex. We interpret these regions in terms of their role in episodic recall, event representation and social-emotional processing. We suggest that different types of abstract concepts can be represented and grounded through differing contributions from event-based, interoceptive, introspective and sensory-motor representations. The results underscore the richness and diversity of abstract concepts, argue against single-mechanism accounts for representation of all types of abstract concepts and suggest mechanisms for their direct and indirect grounding.This article is part of the theme issue 'Varieties of abstract concepts: development, use and representation in the brain'.