One contribution of 23 to a theme issue 'Varieties of abstract concepts: development, use and representation in the brain'. The capacity for abstract thought is one of the hallmarks of human cognition. However, the mechanisms underlying the ability to form and use abstract concepts like 'fantasy' and 'grace' have not been elucidated yet. This theme issue brings together developmental, social and cognitive psychologists, linguists, anthropologists, cognitive scientists, neuroscientists, philosophers and computer scientists to present theoretical insights and novel evidence on how abstract concepts are acquired, used and represented in the brain. Many of the contributions conceive concepts as grounded in sensorimotor systems and constrained by bodily mechanisms and structures. The theme issue develops along two main axes, related to the most promising research directions on abstract concepts. The axes focus on (i) the different kinds of abstract concepts (numbers, emotions, evaluative concepts like moral and aesthetic ones, social concepts); (ii) the role played by perception and action, language and sociality, and inner processes (emotions, interoception, metacognition) in grounding abstract concepts. Most papers adopt a cognitive science/neuroscience approach, but the theme issue also includes studies on development, on social cognition, and on how linguistic diversity shapes abstract concepts. Overall, the theme issue provides an integrated theoretical account that highlights the importance of language, sociality and inner processes for abstract concepts, and that offers new methodological tools to investigate them.This article is part of the theme issue 'Varieties of abstract concepts: development, use and representation in the brain'.