  A new class of Poisson algebras, the class of {\em generalized Weyl Poisson
algebras}, is introduced. It can be seen as Poisson algebra analogue of
generalized Weyl algebras or as giving a Poisson structure to (certain)
generalized Weyl algebras. A Poisson simplicity criterion is given for
generalized Weyl Poisson algebras and explicit descriptions of the Poisson
centre and the absolute Poisson centre are obtained. Many examples are
considered.
