  Using arbitrary symplectic structures and parametrization invariant actions,
we develop a formalism, based on Dirac's quantization procedure, that allows us
to consider theories with both space-space as well as space-time
noncommutativity. Because the formalism has as a starting point an action, the
procedure admits quantizing the theory either by obtaining the quantum
evolution equations or by using the path integral techniques. For both
approaches we only need to select a complete basis of commutative observables.
We show that for certain choices of the potentials that generate a given
symplectic structure, the phase of the quantum transition function between the
admissible bases corresponds to a linear canonical transformation, by means of
which the actions associated to each of these bases may be related and hence
lead to equivalent quantizations. There are however other potentials that
result in actions which can not be related to the previous ones by canonical
transformations, and for which the fixed end-points, in terms of the admissible
bases, can only be realized by means of a Darboux map. In such cases the
original arbitrary symplectic structure is reduced to its canonical form and
therefore each of these actions results in a different quantum theory. One
interesting feature of the formalism here discussed is that it can be
introduced both at the levels of particle systems as well as of field theory.
