  Recently many new classes of integrable systems in n dimensions occurring in
classical and quantum mechanics have been shown to admit a functionally
independent set of 2n-1 symmetries polynomial in the canonical momenta, so that
they are in fact superintegrable. These newly discovered systems are all
separable in some coordinate system and, typically, they depend on one or more
parameters in such a way that the system is superintegrable exactly when some
of the parameters are rational numbers. Most of the constructions to date are
for n=2 but cases where n>2 are multiplying rapidly. In this article we
organize a large class of such systems, many new, and emphasize the underlying
mechanisms which enable this phenomena to occur and to prove
superintegrability. In addition to proofs of classical superintegrability we
show that the 2D caged anisotropic oscillator and a Stackel transformed version
on the 2-sheet hyperboloid are quantum superintegrable for all rational
relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum
superintegrable for all rational values of a parameter k in the potential.
