  A non--commutative analogue of the classical differential forms is
constructed on the phase--space of an arbitrary quantum system. The
non--commutative forms are universal and are related to the quantum mechanical
dynamics in the same way as the classical forms are related to classical
dynamics.
  They are constructed by applying the Weyl--Wigner symbol map to the
differential envelope of the linear operators on the quantum mechanical Hilbert
space. This leads to a representation of the non--commutative forms considered
by A.~Connes in terms of multiscalar functions on the classical phase--space.
In an appropriate coincidence limit they define a quantum deformation of the
classical tensor fields and both commutative and non--commutative forms can be
studied in a unified framework. We interprete the quantum differential forms in
physical terms and comment on possible applications.
