Questions on the nature of concepts in philosophy, psychology, and cognitive science, such as 'What are concepts?' and 'What is it to possess a concept?' are notoriously difficult to answer. For example, are concepts abstract mind-independent objects in some Platonic or Fregean sense, or are they better understood as mental representations, such as constituents of thoughts? A common view in cognitive science is that thought is based on word-like mental representations; some say that possessing a concept C involves demonstrating some kind of ability with respect to C's. But which ability? Other longstanding issues concern a proper theory of the structure of concepts. These questions are tackled here by Simon Baron-Cohen, Peter Carruthers, and a distinguished cast of scientists and philosophers.