Besides its visible rotational symmetries, the Schrodinger equation for the hydrogen atom also has “hidden” symmetries, which reveal themselves as representations of generally noncompact Lie groups in energy eigenspaces. They were discovered by Pauli, who was led to them by quantizing the corresponding symmetries of the classical Kepler problem. In this talk I shall examine Pauli’s hidden symmetries from the point of view of algebraic families of representations, from the point of view of Hilbert space theory, and from the traditional point of view of physics; the relationships between these perspectives are surprisingly intricate. This is joint work with Eyal Subag.