The problem of how concepts can refer to or be about the non-mental world is particularly puzzling for abstract concepts. There is growing evidence that many characteristics beyond the perceptual are involved in grounding different kinds of abstract concept. A resource that has been suggested, but little explored, is introspection. This paper develops that suggestion by focusing specifically on metacognition—on the thoughts and feelings that thinkers have about a concept. One example of metacognition about concepts is the judgement that we should defer to others in how a given concept is used. Another example is our internal assessment of which concepts are dependable and useful, and which less so. Metacognition of this kind may be especially important for grounding abstract concepts. This article is part of the theme issue ‘Varieties of abstract concepts: development, use and representation in the brain’.