The paper outlines one of the most important challenges that embodied and grounded theories need to face, i.e., that to explain how abstract concepts (abstractness) are acquired, represented, and used. I illustrate the view according to which abstract concepts are grounded not only in sensorimotor experiences, like concrete concepts, but also and to a greater extent in linguistic, social, and inner experiences. Specifically, I discuss the role played by metacognition, inner speech, social metacognition, and interoception. I also present evidence showing that the weight of linguistic, social, and inner experiences varies depending on the considered sub-kind of abstract concepts (e.g., mental states and spiritual concepts, numbers, emotions, social concepts). I argue that the challenge to explain abstract concepts representation implies the recognition of: a. the role of language, intended as inner and social tool, in shaping our mind; b. the importance of differences across languages; c. the existence of different kinds of abstract concepts; d. the necessity to adopt new paradigms, able to capture the use of abstract concepts in context and interactive situations. This challenge should be addressed with an integrated approach that bridges developmental, anthropological, and neuroscientific studies. This approach extends embodied and grounded views incorporating insights from distributional statistics views of meaning, from pragmatics and semiotics.