  We propose a new formula for the adiabatic Berry phase which is based on
phase-space formulation of quantum mechanics. This approach sheds a new light
into the correspondence between classical and quantum adiabatic phases -- both
phases are related with the averaging procedure: Hannay's angle with averaging
over the classical torus and Berry's phase with averaging over the entire
classical phase space with respect to the corresponding Wigner function.
Generalizations to the non-abelian Wilczek--Zee case and mixed states are also
included.
