  Phase Space is the framework best suited for quantizing superintegrable
systems--systems with more conserved quantities than degrees of freedom. In
this quantization method, the symmetry algebras of the hamiltonian invariants
are preserved most naturally. We illustrate the power and simplicity of the
method through new applications to nonlinear sigma-models, specifically for
Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians
amount to compact and elegant expressions, in accord with the Groenewold-van
Hove theorem. Additional power and elegance is provided by the use of Nambu
Brackets (linked to Dirac Brackets) involving the extra invariants of
superintegrable models. The quantization of Nambu Brackets is then successfully
compared to that of Moyal, validating Nambu's original proposal, while
invalidating other proposals.
