  We outline an abstract approach to the pseudo-differential Weyl calculus for
operators in function spaces in infinitely many variables. Our earlier approach
to the Weyl calculus for Lie group representations is extended to the case of
representations associated with infinite-dimensional coadjoint orbits. We
illustrate the approach by the case of infinite-dimensional Heisenberg groups.
The classical Weyl-H\"ormander calculus is recovered for the Schr\"odinger
representations of the finite-dimensional Heisenberg groups.
