Only human beings create the conceptual repertoire that underlies the lexicons of natural languages and that subserves the institutions of science, mathematics, government, religion, and art. Accounting for this astonishing feat is one of the foremost challenges in the cognitive and psychological sciences. As a matter of logic, any account must have the following structure: a specification of the initial representational repertoire that serves as the input to subsequent learning, a characterization of the differences between the initial and adult repertoires, and a characterization of the learning mechanisms that make possible conceptual development. In Part I (written by Renée Baillargeon), we first consider the question of the initial repertoire by presenting a case study of the system of representations that underlies reasoning about the physical world. We illustrate the properties of what has been called “core cognition” in the domain of physics, and also characterize how learning underlies changes in this system of representation within infancy. In Part II (written by Susan Carey), we turn to a case study of the acquisition of concepts of number. We characterize the core cognition systems with numerical content, outline the discontinuities between the initial representations and later ones, and describe the bootstrapping process that underlies the change.