  The first part of this paper explains what super-integrability is and how it
differs in the classical and quantum cases. This is illustrated with an
elementary example of the resonant harmonic oscillator.
  For Hamiltonians in "natural form", the kinetic energy has geometric origins
and, in the flat and constant curvature cases, the large isometry group plays a
vital role. We explain how to use the corresponding first integrals to build
separable and super-integrable systems. We also show how to use the
automorphisms of the symmetry algebra to help build the Poisson relations of
the corresponding non-Abelian Poisson algebra.
  Finally, we take both the classical and quantum Zernike system, recently
discussed by Pogosyan, et al, and show how the algebraic structure of its
super-integrability can be understood in this framework.
