  We lay out the foundations of the theory of second-order conformal
superintegrable systems. Such systems are essentially Laplace equations on a
manifold with an added potential: $(\Delta_n+V({\bf x}))\Psi=0$. Distinct
families of second-order superintegrable Schr\"odinger (or Helmholtz) systems
$(\Delta'_n+V'({\bf x}))\Psi=E\Psi$ can be incorporated into a single Laplace
equation. There is a deep connection between most of the special functions of
mathematical physics, these Laplace conformally superintegrable systems and
their conformal symmetry algebras. Using the theory of the Laplace systems, we
show that the problem of classifying all 3D Helmholtz superintegrable systems
with nondegenerate potentials, i.e., potentials with a maximal number of
independent paprameters, can be reduced to the problem of classifying the
orbits of the nonlinear action of the conformal group on a 10-dimensional
manifold.
