The FBI transform on compact C-infinity manifolds

We present a geometric theory of the Fourier-Bros-Iagolnitzer transform on a compact C-infinity manifold M. The FBI transform is a generalization of t he classical notion of the wave-packet transform. We discuss the mapping pr operties of the FBI transform and its relationship to the calculus of pseud odifferential operators on M. We also describe the microlocal properties of its range in terms of the "scattering calculus" of pseudodifferential oper ators on the noncompact manifold T*M.