  This paper presents the general theory of canonical transformations of
coordinates in quantum mechanics. First, the theory is developed in the
formalism of phase space quantum mechanics. It is shown that by transforming a
star-product, when passing to a new coordinate system, observables and states
transform as in classical mechanics, i.e., by composing them with a
transformation of coordinates. Then the developed formalism of coordinate
transformations is transferred to a standard formulation of quantum mechanics.
In addition, the developed theory is illustrated on examples of particular
classes of quantum canonical transformations.
