  A theory of non-unitary-invertible as well as unitary canonical
transformations is formulated in the context of Weyl's phase space
representations. That all quantum canonical transformations without an explicit
$\hbar$ dependence are also classical mechanical and vice versa is demonstrated
in the phase space. Contrary to some earlier results, it is also shown that the
quantum generators and their classical counterparts are identical and
$\hbar$-independent. The latter is a powerful result bringing the theory of
classical canonical transformations and the $\hbar$-independent quantum ones on
an equal footing.
