Summary: We present a study of non-relativistic superintegrable systems whose invariants are quadratic in the momenta. In two dimensions, there exist only two inequivalent classes of such systems. The symmetries responsible for the accidental degeneracies of those problems are investigated and shown to be best described in terms of polynomial algebras. We also determine the quasi-exactly solvable (QES) systems that can be obtained by dimensional reduction from the two- and three-dimensional superintegrable models, establishing in each case the equivalence between the QES Schrödinger equation and the spectral problem associated to a quadratic element in the questions of a Lie algebra.