  Quadratic algebras are generalizations of Lie algebras which include the
symmetry algebras of 2nd order superintegrable systems in 2 dimensions as
special cases. The superintegrable systems are exactly solvable physical
systems in classical and quantum mechanics. Distinct superintegrable systems
and their quadratic algebras can be related by geometric contractions, induced
by B\^ocher contractions of the conformal Lie algebra
${\mathfrak{so}}(4,\mathbb {C})$ to itself. In this paper we give a precise
definition of B\^ocher contractions and show how they can be classified. They
subsume well known contractions of ${\mathfrak{e}}(2,\mathbb {C})$ and
${\mathfrak{so}}(3,\mathbb {C})$ and have important physical and geometric
meanings, such as the derivation of the Askey scheme for obtaining all
hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We
also classify abstract nondegenerate quadratic algebras in terms of an
invariant that we call a canonical form. We describe an algorithm for finding
the canonical form of such algebras. We calculate explicitly all canonical
forms arising from quadratic algebras of 2D nondegenerate superintegrable
systems on constant curvature spaces and Darboux spaces. We further discuss
contraction of quadratic algebras, focusing on those coming from
superintegrable systems.
