  We extend the Wigner-Weyl-Moyal phase-space formulation of quantum mechanics
to general curved configuration spaces. The underlying phase space is based on
the chosen coordinates of the manifold and their canonically conjugate momenta.
The resulting Wigner function displays the axioms of a quasiprobability
distribution, and any Weyl-ordered operator gets associated with the
corresponding phase-space function, even in the absence of continuous
symmetries. The corresponding quantum Liouville equation reduces to the
classical curved space Liouville equation in the semiclassical limit. We
demonstrate the formalism for a point particle moving on two-dimensional
manifolds, such as a paraboloid or the surface of a sphere. The latter
clarifies the treatment of compact coordinate spaces as well as the relation of
the presented phase-space representation to symmetry groups of the
configuration space.
